| Title: | Plane Geometry |
|---|---|
| Description: | An extensive set of plane geometry routines. Provides R6 classes representing triangles, circles, circular arcs, ellipses, elliptical arcs, lines, hyperbolae, and their plot methods. Also provides R6 classes representing transformations: rotations, reflections, homotheties, scalings, general affine transformations, inversions, Möbius transformations. |
| Authors: | Stéphane Laurent |
| Maintainer: | Stéphane Laurent <[email protected]> |
| License: | GPL-3 |
| Version: | 1.6.0 |
| Built: | 2024-10-30 03:33:15 UTC |
| Source: | https://github.com/stla/planegeometry |
An affine map is given by a 2x2 matrix (a linear transformation) and a vector (the "intercept").
Aget or set the matrix A
bget or set the vector b
new()
Create a new Affine object.
Affine$new(A, b)
Athe 2x2 matrix of the affine map
bthe shift vector of the affine map
A new Affine object.
print()
Show instance of an Affine object.
Affine$print(...)
...ignored
Affine$new(rbind(c(3.5,2),c(0,4)), c(-1, 1.25))
get3x3matrix()
The 3x3 matrix representing the affine map.
Affine$get3x3matrix()
inverse()
The inverse affine transformation, if it exists.
Affine$inverse()
compose()
Compose the reference affine map with another affine map.
Affine$compose(transfo, left = TRUE)
transfoan Affine object
leftlogical, whether to compose at left or at right (i.e.
returns f1 o f0 or f0 o f1)
An Affine object.
transform()
Transform a point or several points by the reference affine map.
Affine$transform(M)
Ma point or a two-column matrix of points, one point per row
transformLine()
Transform a line by the reference affine transformation (only for invertible affine maps).
Affine$transformLine(line)
linea Line object
A Line object.
transformEllipse()
Transform an ellipse by the reference affine transformation (only for an invertible affine map). The result is an ellipse.
Affine$transformEllipse(ell)
ellan Ellipse object or a Circle object
An Ellipse object.
clone()
The objects of this class are cloneable with this method.
Affine$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Affine$print` ## ------------------------------------------------ Affine$new(rbind(c(3.5,2),c(0,4)), c(-1, 1.25))## ------------------------------------------------ ## Method `Affine$print` ## ------------------------------------------------ Affine$new(rbind(c(3.5,2),c(0,4)), c(-1, 1.25))
Return the affine transformation which transforms
ell1 to ell2.
AffineMappingEllipse2Ellipse(ell1, ell2)AffineMappingEllipse2Ellipse(ell1, ell2)
ell1, ell2
|
|
An Affine object.
ell1 <- Ellipse$new(c(1,1), 5, 1, 30) ( ell2 <- Ellipse$new(c(4,-1), 3, 2, 50) ) f <- AffineMappingEllipse2Ellipse(ell1, ell2) f$transformEllipse(ell1) # should be ell2ell1 <- Ellipse$new(c(1,1), 5, 1, 30) ( ell2 <- Ellipse$new(c(4,-1), 3, 2, 50) ) f <- AffineMappingEllipse2Ellipse(ell1, ell2) f$transformEllipse(ell1) # should be ell2
Return the affine transformation which sends
P1 to Q1, P2 to Q2 and P3 to Q3.
AffineMappingThreePoints(P1, P2, P3, Q1, Q2, Q3)AffineMappingThreePoints(P1, P2, P3, Q1, Q2, Q3)
P1, P2, P3
|
three non-collinear points |
Q1, Q2, Q3
|
three non-collinear points |
An Affine object.
An arc is given by a center, a radius, a starting angle and
an ending angle. They are respectively named center, radius,
alpha1 and alpha2.
centerget or set the center
radiusget or set the radius
alpha1get or set the starting angle
alpha2get or set the ending angle
degreesget or set the degrees field
new()
Create a new Arc object.
Arc$new(center, radius, alpha1, alpha2, degrees = TRUE)
centerthe center
radiusthe radius
alpha1the starting angle
alpha2the ending angle
degreeslogical, whether alpha1 and alpha2 are
given in degrees
A new Arc object.
arc <- Arc$new(c(1,1), 1, 45, 90) arc arc$center arc$center <- c(0,0) arc
print()
Show instance of an Arc object.
Arc$print(...)
...ignored
Arc$new(c(0,0), 2, pi/4, pi/2, FALSE)
startingPoint()
Starting point of the reference arc.
Arc$startingPoint()
endingPoint()
Ending point of the reference arc.
Arc$endingPoint()
isEqual()
Check whether the reference arc equals another arc.
Arc$isEqual(arc)
arcan Arc object
complementaryArc()
Complementary arc of the reference arc.
Arc$complementaryArc()
arc <- Arc$new(c(0,0), 1, 30, 60)
plot(NULL, type = "n", asp = 1, xlim = c(-1,1), ylim = c(-1,1),
xlab = NA, ylab = NA)
draw(arc, lwd = 3, col = "red")
draw(arc$complementaryArc(), lwd = 3, col = "green")
path()
The reference arc as a path.
Arc$path(npoints = 100L)
npointsnumber of points of the path
A matrix with two columns x and y of length
npoints. See "Filling the lapping area of two circles" in the
vignette for an example.
clone()
The objects of this class are cloneable with this method.
Arc$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Arc$new` ## ------------------------------------------------ arc <- Arc$new(c(1,1), 1, 45, 90) arc arc$center arc$center <- c(0,0) arc ## ------------------------------------------------ ## Method `Arc$print` ## ------------------------------------------------ Arc$new(c(0,0), 2, pi/4, pi/2, FALSE) ## ------------------------------------------------ ## Method `Arc$complementaryArc` ## ------------------------------------------------ arc <- Arc$new(c(0,0), 1, 30, 60) plot(NULL, type = "n", asp = 1, xlim = c(-1,1), ylim = c(-1,1), xlab = NA, ylab = NA) draw(arc, lwd = 3, col = "red") draw(arc$complementaryArc(), lwd = 3, col = "green")## ------------------------------------------------ ## Method `Arc$new` ## ------------------------------------------------ arc <- Arc$new(c(1,1), 1, 45, 90) arc arc$center arc$center <- c(0,0) arc ## ------------------------------------------------ ## Method `Arc$print` ## ------------------------------------------------ Arc$new(c(0,0), 2, pi/4, pi/2, FALSE) ## ------------------------------------------------ ## Method `Arc$complementaryArc` ## ------------------------------------------------ arc <- Arc$new(c(0,0), 1, 30, 60) plot(NULL, type = "n", asp = 1, xlim = c(-1,1), ylim = c(-1,1), xlab = NA, ylab = NA) draw(arc, lwd = 3, col = "red") draw(arc$complementaryArc(), lwd = 3, col = "green")
A circle is given by a center and a radius,
named center and radius.
centerget or set the center
radiusget or set the radius
new()
Create a new Circle object.
Circle$new(center, radius)
centerthe center
radiusthe radius
A new Circle object.
circ <- Circle$new(c(1,1), 1) circ circ$center circ$center <- c(0,0) circ
print()
Show instance of a circle object.
Circle$print(...)
...ignored
Circle$new(c(0,0), 2)
pointFromAngle()
Get a point on the reference circle from its polar angle.
Circle$pointFromAngle(alpha, degrees = TRUE)
alphaa number, the angle
degreeslogical, whether alpha is given in degrees
The point on the circle with polar angle alpha.
diameter()
Diameter of the reference circle for a given polar angle.
Circle$diameter(alpha)
alphaan angle in radians, there is one diameter for each value of
alpha modulo pi
A segment (Line object).
circ <- Circle$new(c(1,1), 5)
diams <- lapply(c(0, pi/3, 2*pi/3), circ$diameter)
plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7),
xlab = NA, ylab = NA)
draw(circ, lwd = 2, col = "yellow")
invisible(lapply(diams, draw, col = "blue"))
tangent()
Tangent of the reference circle at a given polar angle.
Circle$tangent(alpha)
alphaan angle in radians, there is one tangent for each value of
alpha modulo 2*pi
circ <- Circle$new(c(1,1), 5)
tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), circ$tangent)
plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7),
xlab = NA, ylab = NA)
draw(circ, lwd = 2, col = "yellow")
invisible(lapply(tangents, draw, col = "blue"))
tangentsThroughExternalPoint()
Return the two tangents of the reference circle passing through an external point.
Circle$tangentsThroughExternalPoint(P)
Pa point external to the reference circle
A list of two Line objects, the two tangents; the
tangency points are in the B field of the lines.
isEqual()
Check whether the reference circle equals another circle.
Circle$isEqual(circ)
circa Circle object
isDifferent()
Check whether the reference circle differs from another circle.
Circle$isDifferent(circ)
circa Circle object
isOrthogonal()
Check whether the reference circle is orthogonal to a given circle.
Circle$isOrthogonal(circ)
circa Circle object
angle()
Angle between the reference circle and a given circle, if they intersect.
Circle$angle(circ)
circa Circle object
includes()
Check whether a point belongs to the reference circle.
Circle$includes(M)
Ma point
orthogonalThroughTwoPointsOnCircle()
Orthogonal circle passing through two points on the reference circle.
Circle$orthogonalThroughTwoPointsOnCircle(alpha1, alpha2, arc = FALSE)
alpha1, alpha2two angles defining two points on the reference circle
arclogical, whether to return only the arc at the interior of the reference circle
A Circle object if arc=FALSE, an Arc object
if arc=TRUE, or a Line object: the diameter
of the reference circle defined by the two points in case when the two
angles differ by pi.
# hyperbolic triangle circ <- Circle$new(c(5,5), 3) arc1 <- circ$orthogonalThroughTwoPointsOnCircle(0, 2*pi/3, arc = TRUE) arc2 <- circ$orthogonalThroughTwoPointsOnCircle(2*pi/3, 4*pi/3, arc = TRUE) arc3 <- circ$orthogonalThroughTwoPointsOnCircle(4*pi/3, 0, arc = TRUE) opar <- par(mar = c(0,0,0,0)) plot(0, 0, type = "n", asp = 1, xlim = c(2,8), ylim = c(2,8)) draw(circ) draw(arc1, col = "red", lwd = 2) draw(arc2, col = "green", lwd = 2) draw(arc3, col = "blue", lwd = 2) par(opar)
orthogonalThroughTwoPointsWithinCircle()
Orthogonal circle passing through two points within the reference circle.
Circle$orthogonalThroughTwoPointsWithinCircle(P1, P2, arc = FALSE)
P1, P2two distinct points in the interior of the reference circle
arclogical, whether to return the arc joining the two points instead of the circle
A Circle object or an Arc object,
or a Line object if the two points are on a diameter.
circ <- Circle$new(c(0,0),3)
P1 <- c(1,1); P2 <- c(1, 2)
ocirc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2)
arc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2, arc = TRUE)
plot(0, 0, type = "n", asp = 1, xlab = NA, ylab = NA,
xlim = c(-3, 4), ylim = c(-3, 4))
draw(circ, lwd = 2)
draw(ocirc, lty = "dashed", lwd = 2)
draw(arc, lwd = 3, col = "blue")
power()
Power of a point with respect to the reference circle.
Circle$power(M)
Mpoint
A number.
radicalCenter()
Radical center of two circles.
Circle$radicalCenter(circ2)
circ2a Circle object
radicalAxis()
Radical axis of two circles.
Circle$radicalAxis(circ2)
circ2a Circle object
A Line object.
rotate()
Rotate the reference circle.
Circle$rotate(alpha, O, degrees = TRUE)
alphaangle of rotation
Ocenter of rotation
degreeslogical, whether alpha is given in degrees
A Circle object.
translate()
Translate the reference circle.
Circle$translate(v)
vthe vector of translation
A Circle object.
invert()
Invert the reference circle.
Circle$invert(inversion)
inversionan Inversion object
A Circle object or a Line object.
asEllipse()
Convert the reference circle to an Ellipse object.
Circle$asEllipse()
randomPoints()
Random points on or in the reference circle.
Circle$randomPoints(n, where = "in")
nan integer, the desired number of points
where"in" to generate inside the circle,
"on" to generate on the circle
The generated points in a two columns matrix with n rows.
clone()
The objects of this class are cloneable with this method.
Circle$clone(deep = FALSE)
deepWhether to make a deep clone.
radicalCenter for the radical center of three circles.
## ------------------------------------------------ ## Method `Circle$new` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 1) circ circ$center circ$center <- c(0,0) circ ## ------------------------------------------------ ## Method `Circle$print` ## ------------------------------------------------ Circle$new(c(0,0), 2) ## ------------------------------------------------ ## Method `Circle$diameter` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 5) diams <- lapply(c(0, pi/3, 2*pi/3), circ$diameter) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7), xlab = NA, ylab = NA) draw(circ, lwd = 2, col = "yellow") invisible(lapply(diams, draw, col = "blue")) ## ------------------------------------------------ ## Method `Circle$tangent` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 5) tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), circ$tangent) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7), xlab = NA, ylab = NA) draw(circ, lwd = 2, col = "yellow") invisible(lapply(tangents, draw, col = "blue")) ## ------------------------------------------------ ## Method `Circle$orthogonalThroughTwoPointsOnCircle` ## ------------------------------------------------ # hyperbolic triangle circ <- Circle$new(c(5,5), 3) arc1 <- circ$orthogonalThroughTwoPointsOnCircle(0, 2*pi/3, arc = TRUE) arc2 <- circ$orthogonalThroughTwoPointsOnCircle(2*pi/3, 4*pi/3, arc = TRUE) arc3 <- circ$orthogonalThroughTwoPointsOnCircle(4*pi/3, 0, arc = TRUE) opar <- par(mar = c(0,0,0,0)) plot(0, 0, type = "n", asp = 1, xlim = c(2,8), ylim = c(2,8)) draw(circ) draw(arc1, col = "red", lwd = 2) draw(arc2, col = "green", lwd = 2) draw(arc3, col = "blue", lwd = 2) par(opar) ## ------------------------------------------------ ## Method `Circle$orthogonalThroughTwoPointsWithinCircle` ## ------------------------------------------------ circ <- Circle$new(c(0,0),3) P1 <- c(1,1); P2 <- c(1, 2) ocirc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2) arc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2, arc = TRUE) plot(0, 0, type = "n", asp = 1, xlab = NA, ylab = NA, xlim = c(-3, 4), ylim = c(-3, 4)) draw(circ, lwd = 2) draw(ocirc, lty = "dashed", lwd = 2) draw(arc, lwd = 3, col = "blue")## ------------------------------------------------ ## Method `Circle$new` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 1) circ circ$center circ$center <- c(0,0) circ ## ------------------------------------------------ ## Method `Circle$print` ## ------------------------------------------------ Circle$new(c(0,0), 2) ## ------------------------------------------------ ## Method `Circle$diameter` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 5) diams <- lapply(c(0, pi/3, 2*pi/3), circ$diameter) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7), xlab = NA, ylab = NA) draw(circ, lwd = 2, col = "yellow") invisible(lapply(diams, draw, col = "blue")) ## ------------------------------------------------ ## Method `Circle$tangent` ## ------------------------------------------------ circ <- Circle$new(c(1,1), 5) tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), circ$tangent) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-5,7), xlab = NA, ylab = NA) draw(circ, lwd = 2, col = "yellow") invisible(lapply(tangents, draw, col = "blue")) ## ------------------------------------------------ ## Method `Circle$orthogonalThroughTwoPointsOnCircle` ## ------------------------------------------------ # hyperbolic triangle circ <- Circle$new(c(5,5), 3) arc1 <- circ$orthogonalThroughTwoPointsOnCircle(0, 2*pi/3, arc = TRUE) arc2 <- circ$orthogonalThroughTwoPointsOnCircle(2*pi/3, 4*pi/3, arc = TRUE) arc3 <- circ$orthogonalThroughTwoPointsOnCircle(4*pi/3, 0, arc = TRUE) opar <- par(mar = c(0,0,0,0)) plot(0, 0, type = "n", asp = 1, xlim = c(2,8), ylim = c(2,8)) draw(circ) draw(arc1, col = "red", lwd = 2) draw(arc2, col = "green", lwd = 2) draw(arc3, col = "blue", lwd = 2) par(opar) ## ------------------------------------------------ ## Method `Circle$orthogonalThroughTwoPointsWithinCircle` ## ------------------------------------------------ circ <- Circle$new(c(0,0),3) P1 <- c(1,1); P2 <- c(1, 2) ocirc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2) arc <- circ$orthogonalThroughTwoPointsWithinCircle(P1, P2, arc = TRUE) plot(0, 0, type = "n", asp = 1, xlab = NA, ylab = NA, xlim = c(-3, 4), ylim = c(-3, 4)) draw(circ, lwd = 2) draw(ocirc, lty = "dashed", lwd = 2) draw(arc, lwd = 3, col = "blue")
Return the circle given by a diameter
CircleAB(A, B)CircleAB(A, B)
A, B
|
the endpoints of the diameter |
A Circle object.
Return the circle given by its center and a point it passes through.
CircleOA(O, A)CircleOA(O, A)
O |
the center of the circle |
A |
a point of the circle |
A Circle object.
The cross ratio of four points.
crossRatio(A, B, C, D)crossRatio(A, B, C, D)
A, B, C, D
|
four distinct points |
A complex number. It is real if and only if the four points lie on a generalized circle (that is a circle or a line).
c <- Circle$new(c(0, 0), 1) A <- c$pointFromAngle(0) B <- c$pointFromAngle(90) C <- c$pointFromAngle(180) D <- c$pointFromAngle(270) crossRatio(A, B, C, D) # should be real Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) MA <- Mob$transform(A) MB <- Mob$transform(B) MC <- Mob$transform(C) MD <- Mob$transform(D) crossRatio(MA, MB, MC, MD) # should be identical to `crossRatio(A, B, C, D)`c <- Circle$new(c(0, 0), 1) A <- c$pointFromAngle(0) B <- c$pointFromAngle(90) C <- c$pointFromAngle(180) D <- c$pointFromAngle(270) crossRatio(A, B, C, D) # should be real Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) MA <- Mob$transform(A) MB <- Mob$transform(B) MC <- Mob$transform(C) MD <- Mob$transform(D) crossRatio(MA, MB, MC, MD) # should be identical to `crossRatio(A, B, C, D)`
Draw a geometric object on the current plot.
draw(x, ...) ## S3 method for class 'Triangle' draw(x, ...) ## S3 method for class 'Circle' draw(x, npoints = 100L, ...) ## S3 method for class 'Arc' draw(x, npoints = 100L, ...) ## S3 method for class 'Ellipse' draw(x, npoints = 100L, ...) ## S3 method for class 'EllipticalArc' draw(x, npoints = 100L, ...) ## S3 method for class 'Line' draw(x, ...)draw(x, ...) ## S3 method for class 'Triangle' draw(x, ...) ## S3 method for class 'Circle' draw(x, npoints = 100L, ...) ## S3 method for class 'Arc' draw(x, npoints = 100L, ...) ## S3 method for class 'Ellipse' draw(x, npoints = 100L, ...) ## S3 method for class 'EllipticalArc' draw(x, npoints = 100L, ...) ## S3 method for class 'Line' draw(x, ...)
x |
geometric object ( |
... |
arguments passed to |
npoints |
integer, the number of points of the path |
# open new plot window plot(0, 0, type="n", asp = 1, xlim = c(0,2.5), ylim = c(0,2.5), xlab = NA, ylab = NA) grid() # draw a triangle t <- Triangle$new(c(0,0), c(1,0), c(0.5,sqrt(3)/2)) draw(t, col = "blue", lwd = 2) draw(t$rotate(90, t$C), col = "green", lwd = 2) # draw a circle circ <- t$incircle() draw(circ, col = "orange", border = "brown", lwd = 2) # draw an ellipse S <- Scaling$new(circ$center, direction = c(2,1), scale = 2) draw(S$scaleCircle(circ), border = "grey", lwd = 2) # draw a line l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) draw(l, col = "red", lwd = 2) perp <- l$perpendicular(c(2,1)) draw(perp, col = "yellow", lwd = 2)# open new plot window plot(0, 0, type="n", asp = 1, xlim = c(0,2.5), ylim = c(0,2.5), xlab = NA, ylab = NA) grid() # draw a triangle t <- Triangle$new(c(0,0), c(1,0), c(0.5,sqrt(3)/2)) draw(t, col = "blue", lwd = 2) draw(t$rotate(90, t$C), col = "green", lwd = 2) # draw a circle circ <- t$incircle() draw(circ, col = "orange", border = "brown", lwd = 2) # draw an ellipse S <- Scaling$new(circ$center, direction = c(2,1), scale = 2) draw(S$scaleCircle(circ), border = "grey", lwd = 2) # draw a line l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) draw(l, col = "red", lwd = 2) perp <- l$perpendicular(c(2,1)) draw(perp, col = "yellow", lwd = 2)
An ellipse is given by a center, two radii (rmajor
and rminor), and the angle (alpha) between the major axis and
the horizontal direction.
centerget or set the center
rmajorget or set the major radius of the ellipse
rminorget or set the minor radius of the ellipse
alphaget or set the angle of the ellipse
degreesget or set the degrees field
new()
Create a new Ellipse object.
Ellipse$new(center, rmajor, rminor, alpha, degrees = TRUE)
centera point, the center of the rotation
rmajorpositive number, the major radius
rminorpositive number, the minor radius
alphaa number, the angle between the major axis and the horizontal direction
degreeslogical, whether alpha is given in degrees
A new Ellipse object.
Ellipse$new(c(1,1), 3, 2, 30)
print()
Show instance of an Ellipse object.
Ellipse$print(...)
...ignored
isEqual()
Check whether the reference ellipse equals an ellipse.
Ellipse$isEqual(ell)
ellAn Ellipse object.
equation()
The coefficients of the implicit equation of the ellipse.
Ellipse$equation()
The implicit equation of the ellipse is
Ax² + Bxy + Cy² + Dx + Ey + F = 0. This method returns
A, B, C, D, E and F.
A named numeric vector.
includes()
Check whether a point lies on the reference ellipse.
Ellipse$includes(M)
Ma point
contains()
Check whether a point is contained in the reference ellipse.
Ellipse$contains(M)
Ma point
matrix()
Returns the 2x2 matrix S associated to the reference
ellipse. The equation of the ellipse is t(M-O) %*% S %*% (M-O) = 1.
Ellipse$matrix()
ell <- Ellipse$new(c(1,1), 5, 1, 30) S <- ell$matrix() O <- ell$center pts <- ell$path(4L) # four points on the ellipse apply(pts, 1L, function(M) t(M-O) %*% S %*% (M-O))
path()
Path that forms the reference ellipse.
Ellipse$path(npoints = 100L, closed = FALSE, outer = FALSE)
npointsnumber of points of the path
closedBoolean, whether to return a closed path; you don't need
a closed path if you want to plot it with
polygon
outerBoolean; if TRUE, the ellipse will be contained
inside the path, otherwise it will contain the path
A matrix with two columns x and y of
length npoints.
library(PlaneGeometry) ell <- Ellipse$new(c(1, -1), rmajor = 3, rminor = 2, alpha = 30) innerPath <- ell$path(npoints = 10) outerPath <- ell$path(npoints = 10, outer = TRUE) bbox <- ell$boundingbox() plot(NULL, asp = 1, xlim = bbox$x, ylim = bbox$y, xlab = NA, ylab = NA) draw(ell, border = "red", lty = "dashed") polygon(innerPath, border = "blue", lwd = 2) polygon(outerPath, border = "green", lwd = 2)
diameter()
Diameter and conjugate diameter of the reference ellipse.
Ellipse$diameter(t, conjugate = FALSE)
ta number, the diameter only depends on t modulo
pi; the axes correspond to t=0 and t=pi/2
conjugatelogical, whether to return the conjugate diameter as well
A Line object or a list of two Line objects if
conjugate = TRUE.
ell <- Ellipse$new(c(1,1), 5, 2, 30)
diameters <- lapply(c(0, pi/3, 2*pi/3), ell$diameter)
plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4),
xlab = NA, ylab = NA)
draw(ell)
invisible(lapply(diameters, draw))
perimeter()
Perimeter of the reference ellipse.
Ellipse$perimeter()
pointFromAngle()
Intersection point of the ellipse with the half-line
starting at the ellipse center and forming angle theta with
the major axis.
Ellipse$pointFromAngle(theta, degrees = TRUE)
thetaa number, the angle, or a numeric vector
degreeslogical, whether theta is given in degrees
A point of the ellipse if length(theta)==1 or a
two-column matrix of points of the ellipse if
length(theta) > 1 (one point per row).
pointFromEccentricAngle()
Point of the ellipse with given eccentric angle.
Ellipse$pointFromEccentricAngle(t)
ta number, the eccentric angle in radians, or a numeric vector
A point of the ellipse if length(t)==1 or a
two-column matrix of points of the ellipse if
length(t) > 1 (one point per row).
semiMajorAxis()
Semi-major axis of the ellipse.
Ellipse$semiMajorAxis()
A segment (Line object).
semiMinorAxis()
Semi-minor axis of the ellipse.
Ellipse$semiMinorAxis()
A segment (Line object).
foci()
Foci of the reference ellipse.
Ellipse$foci()
A list with the two foci.
tangent()
Tangents of the reference ellipse at a point given by its eccentric angle.
Ellipse$tangent(t)
teccentric angle, there is one tangent for each value of t
modulo 2*pi; for t = 0, pi/2, pi, -pi/2, these are the
tangents at the vertices of the ellipse
ell <- Ellipse$new(c(1,1), 5, 2, 30)
tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), ell$tangent)
plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4),
xlab = NA, ylab = NA)
draw(ell, col = "yellow")
invisible(lapply(tangents, draw, col = "blue"))
normal()
Normal unit vector to the ellipse.
Ellipse$normal(t)
ta number, the eccentric angle in radians of the point of the ellipse at which we want the normal unit vector
The normal unit vector to the ellipse at the point given by
eccentric angle t.
ell <- Ellipse$new(c(1,1), 5, 2, 30)
t_ <- seq(0, 2*pi, length.out = 13)[-1]
plot(NULL, asp = 1, xlim = c(-5,7), ylim = c(-3,5),
xlab = NA, ylab = NA)
draw(ell, col = "magenta")
for(i in 1:length(t_)){
t <- t_[i]
P <- ell$pointFromEccentricAngle(t)
v <- ell$normal(t)
draw(Line$new(P, P+v, FALSE, FALSE))
}
theta2t()
Convert angle to eccentric angle.
Ellipse$theta2t(theta, degrees = TRUE)
thetaangle between the major axis and the half-line starting at the center of the ellipse and passing through the point of interest on the ellipse
degreeslogical, whether theta is given in degrees
The eccentric angle of the point of interest on the ellipse, in radians.
O <- c(1, 1)
ell <- Ellipse$new(O, 5, 2, 30)
theta <- 20
P <- ell$pointFromAngle(theta)
t <- ell$theta2t(theta)
tg <- ell$tangent(t)
OP <- Line$new(O, P, FALSE, FALSE)
plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,5),
xlab = NA, ylab = NA)
draw(ell, col = "antiquewhite")
points(P[1], P[2], pch = 19)
draw(tg, col = "red")
draw(OP)
draw(ell$semiMajorAxis())
text(t(O+c(1,0.9)), expression(theta))
regressionLines()
Regression lines. The regression line of y on x intersects the ellipse at its rightmost point and its leftmost point. The tangents at these points are vertical. The regression line of x on y intersects the ellipse at its topmost point and its bottommost point. The tangents at these points are horizontal.
Ellipse$regressionLines()
A list with two Line objects:
the regression line of y on x and the regression line of x on y.
ell <- Ellipse$new(c(1,1), 5, 2, 30)
reglines <- ell$regressionLines()
plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4),
xlab = NA, ylab = NA)
draw(ell, lwd = 2)
draw(reglines$YonX, lwd = 2, col = "blue")
draw(reglines$XonY, lwd = 2, col = "green")
boundingbox()
Return the smallest rectangle parallel to the axes which contains the reference ellipse.
Ellipse$boundingbox()
A list with two components: the x-limits in x and the
y-limits in y.
ell <- Ellipse$new(c(2,2), 5, 3, 40) box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, col = "seaShell", border = "blue") abline(v = box$x, lty = 2); abline(h = box$y, lty = 2)
randomPoints()
Random points on or in the reference ellipse.
Ellipse$randomPoints(n, where = "in")
nan integer, the desired number of points
where"in" to generate inside the ellipse,
"on" to generate on the ellipse
The generated points in a two columns matrix with n rows.
ell <- Ellipse$new(c(1,1), 5, 2, 30)
pts <- ell$randomPoints(100)
plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-2,4),
xlab = NA, ylab = NA)
draw(ell, lwd = 2)
points(pts, pch = 19, col = "blue")
clone()
The objects of this class are cloneable with this method.
Ellipse$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Ellipse$new` ## ------------------------------------------------ Ellipse$new(c(1,1), 3, 2, 30) ## ------------------------------------------------ ## Method `Ellipse$matrix` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 1, 30) S <- ell$matrix() O <- ell$center pts <- ell$path(4L) # four points on the ellipse apply(pts, 1L, function(M) t(M-O) %*% S %*% (M-O)) ## ------------------------------------------------ ## Method `Ellipse$path` ## ------------------------------------------------ library(PlaneGeometry) ell <- Ellipse$new(c(1, -1), rmajor = 3, rminor = 2, alpha = 30) innerPath <- ell$path(npoints = 10) outerPath <- ell$path(npoints = 10, outer = TRUE) bbox <- ell$boundingbox() plot(NULL, asp = 1, xlim = bbox$x, ylim = bbox$y, xlab = NA, ylab = NA) draw(ell, border = "red", lty = "dashed") polygon(innerPath, border = "blue", lwd = 2) polygon(outerPath, border = "green", lwd = 2) ## ------------------------------------------------ ## Method `Ellipse$diameter` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) diameters <- lapply(c(0, pi/3, 2*pi/3), ell$diameter) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell) invisible(lapply(diameters, draw)) ## ------------------------------------------------ ## Method `Ellipse$tangent` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), ell$tangent) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, col = "yellow") invisible(lapply(tangents, draw, col = "blue")) ## ------------------------------------------------ ## Method `Ellipse$normal` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) t_ <- seq(0, 2*pi, length.out = 13)[-1] plot(NULL, asp = 1, xlim = c(-5,7), ylim = c(-3,5), xlab = NA, ylab = NA) draw(ell, col = "magenta") for(i in 1:length(t_)){ t <- t_[i] P <- ell$pointFromEccentricAngle(t) v <- ell$normal(t) draw(Line$new(P, P+v, FALSE, FALSE)) } ## ------------------------------------------------ ## Method `Ellipse$theta2t` ## ------------------------------------------------ O <- c(1, 1) ell <- Ellipse$new(O, 5, 2, 30) theta <- 20 P <- ell$pointFromAngle(theta) t <- ell$theta2t(theta) tg <- ell$tangent(t) OP <- Line$new(O, P, FALSE, FALSE) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,5), xlab = NA, ylab = NA) draw(ell, col = "antiquewhite") points(P[1], P[2], pch = 19) draw(tg, col = "red") draw(OP) draw(ell$semiMajorAxis()) text(t(O+c(1,0.9)), expression(theta)) ## ------------------------------------------------ ## Method `Ellipse$regressionLines` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) reglines <- ell$regressionLines() plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, lwd = 2) draw(reglines$YonX, lwd = 2, col = "blue") draw(reglines$XonY, lwd = 2, col = "green") ## ------------------------------------------------ ## Method `Ellipse$boundingbox` ## ------------------------------------------------ ell <- Ellipse$new(c(2,2), 5, 3, 40) box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, col = "seaShell", border = "blue") abline(v = box$x, lty = 2); abline(h = box$y, lty = 2) ## ------------------------------------------------ ## Method `Ellipse$randomPoints` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) pts <- ell$randomPoints(100) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, lwd = 2) points(pts, pch = 19, col = "blue")## ------------------------------------------------ ## Method `Ellipse$new` ## ------------------------------------------------ Ellipse$new(c(1,1), 3, 2, 30) ## ------------------------------------------------ ## Method `Ellipse$matrix` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 1, 30) S <- ell$matrix() O <- ell$center pts <- ell$path(4L) # four points on the ellipse apply(pts, 1L, function(M) t(M-O) %*% S %*% (M-O)) ## ------------------------------------------------ ## Method `Ellipse$path` ## ------------------------------------------------ library(PlaneGeometry) ell <- Ellipse$new(c(1, -1), rmajor = 3, rminor = 2, alpha = 30) innerPath <- ell$path(npoints = 10) outerPath <- ell$path(npoints = 10, outer = TRUE) bbox <- ell$boundingbox() plot(NULL, asp = 1, xlim = bbox$x, ylim = bbox$y, xlab = NA, ylab = NA) draw(ell, border = "red", lty = "dashed") polygon(innerPath, border = "blue", lwd = 2) polygon(outerPath, border = "green", lwd = 2) ## ------------------------------------------------ ## Method `Ellipse$diameter` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) diameters <- lapply(c(0, pi/3, 2*pi/3), ell$diameter) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell) invisible(lapply(diameters, draw)) ## ------------------------------------------------ ## Method `Ellipse$tangent` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) tangents <- lapply(c(0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3), ell$tangent) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, col = "yellow") invisible(lapply(tangents, draw, col = "blue")) ## ------------------------------------------------ ## Method `Ellipse$normal` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) t_ <- seq(0, 2*pi, length.out = 13)[-1] plot(NULL, asp = 1, xlim = c(-5,7), ylim = c(-3,5), xlab = NA, ylab = NA) draw(ell, col = "magenta") for(i in 1:length(t_)){ t <- t_[i] P <- ell$pointFromEccentricAngle(t) v <- ell$normal(t) draw(Line$new(P, P+v, FALSE, FALSE)) } ## ------------------------------------------------ ## Method `Ellipse$theta2t` ## ------------------------------------------------ O <- c(1, 1) ell <- Ellipse$new(O, 5, 2, 30) theta <- 20 P <- ell$pointFromAngle(theta) t <- ell$theta2t(theta) tg <- ell$tangent(t) OP <- Line$new(O, P, FALSE, FALSE) plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,5), xlab = NA, ylab = NA) draw(ell, col = "antiquewhite") points(P[1], P[2], pch = 19) draw(tg, col = "red") draw(OP) draw(ell$semiMajorAxis()) text(t(O+c(1,0.9)), expression(theta)) ## ------------------------------------------------ ## Method `Ellipse$regressionLines` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) reglines <- ell$regressionLines() plot(NULL, asp = 1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, lwd = 2) draw(reglines$YonX, lwd = 2, col = "blue") draw(reglines$XonY, lwd = 2, col = "green") ## ------------------------------------------------ ## Method `Ellipse$boundingbox` ## ------------------------------------------------ ell <- Ellipse$new(c(2,2), 5, 3, 40) box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, col = "seaShell", border = "blue") abline(v = box$x, lty = 2); abline(h = box$y, lty = 2) ## ------------------------------------------------ ## Method `Ellipse$randomPoints` ## ------------------------------------------------ ell <- Ellipse$new(c(1,1), 5, 2, 30) pts <- ell$randomPoints(100) plot(NULL, type="n", asp=1, xlim = c(-4,6), ylim = c(-2,4), xlab = NA, ylab = NA) draw(ell, lwd = 2) points(pts, pch = 19, col = "blue")
The coefficients of the implicit equation of an ellipse from five points on this ellipse.
EllipseEquationFromFivePoints(P1, P2, P3, P4, P5)EllipseEquationFromFivePoints(P1, P2, P3, P4, P5)
P1, P2, P3, P4, P5
|
the five points |
The implicit equation of the ellipse is
Ax² + Bxy + Cy² + Dx + Ey + F = 0. This function returns
A, B, C, D, E and F.
A named numeric vector.
ell <- Ellipse$new(c(2,3), 5, 4, 30) set.seed(666) pts <- ell$randomPoints(5, "on") cf1 <- EllipseEquationFromFivePoints(pts[1,],pts[2,],pts[3,],pts[4,],pts[5,]) cf2 <- ell$equation() # should be the same up to a multiplicative factor all.equal(cf1/cf1["F"], cf2/cf2["F"])ell <- Ellipse$new(c(2,3), 5, 4, 30) set.seed(666) pts <- ell$randomPoints(5, "on") cf1 <- EllipseEquationFromFivePoints(pts[1,],pts[2,],pts[3,],pts[4,],pts[5,]) cf2 <- ell$equation() # should be the same up to a multiplicative factor all.equal(cf1/cf1["F"], cf2/cf2["F"])
Returns the ellipse of equation
t(X-center) %*% S %*% (X-center) = 1.
EllipseFromCenterAndMatrix(center, S)EllipseFromCenterAndMatrix(center, S)
center |
a point, the center of the ellipse |
S |
a positive symmetric matrix |
An Ellipse object.
ell <- Ellipse$new(c(2,3), 4, 2, 20) S <- ell$matrix() EllipseFromCenterAndMatrix(ell$center, S)ell <- Ellipse$new(c(2,3), 4, 2, 20) S <- ell$matrix() EllipseFromCenterAndMatrix(ell$center, S)
Return an ellipse from the coefficients of its implicit equation.
EllipseFromEquation(A, B, C, D, E, F)EllipseFromEquation(A, B, C, D, E, F)
A, B, C, D, E, F
|
the coefficients of the equation |
The implicit equation of the ellipse is
Ax² + Bxy + Cy² + Dx + Ey + F = 0. This function returns the ellipse
given A, B, C, D, E and F.
An Ellipse object.
ell <- Ellipse$new(c(2,3), 5, 4, 30) cf <- ell$equation() ell2 <- EllipseFromEquation(cf[1], cf[2], cf[3], cf[4], cf[5], cf[6]) ell$isEqual(ell2)ell <- Ellipse$new(c(2,3), 5, 4, 30) cf <- ell$equation() ell2 <- EllipseFromEquation(cf[1], cf[2], cf[3], cf[4], cf[5], cf[6]) ell$isEqual(ell2)
Return an ellipse from five given points on this ellipse.
EllipseFromFivePoints(P1, P2, P3, P4, P5)EllipseFromFivePoints(P1, P2, P3, P4, P5)
P1, P2, P3, P4, P5
|
the five points |
An Ellipse object.
ell <- Ellipse$new(c(2,3), 5, 4, 30) set.seed(666) pts <- ell$randomPoints(5, "on") ell2 <- EllipseFromFivePoints(pts[1,],pts[2,],pts[3,],pts[4,],pts[5,]) ell$isEqual(ell2)ell <- Ellipse$new(c(2,3), 5, 4, 30) set.seed(666) pts <- ell$randomPoints(5, "on") ell2 <- EllipseFromFivePoints(pts[1,],pts[2,],pts[3,],pts[4,],pts[5,]) ell$isEqual(ell2)
Derive the ellipse with given foci and one point on the boundary.
EllipseFromFociAndOnePoint(F1, F2, P)EllipseFromFociAndOnePoint(F1, F2, P)
F1, F2
|
points, the foci |
P |
a point on the boundary of the ellipse |
An Ellipse object.
Returns the smallest area ellipse which passes through three given boundary points.
EllipseFromThreeBoundaryPoints(P1, P2, P3)EllipseFromThreeBoundaryPoints(P1, P2, P3)
P1, P2, P3
|
three non-collinear points |
An Ellipse object.
P1 <- c(-1,0); P2 <- c(0, 2); P3 <- c(3,0) ell <- EllipseFromThreeBoundaryPoints(P1, P2, P3) ell$includes(P1); ell$includes(P2); ell$includes(P3)P1 <- c(-1,0); P2 <- c(0, 2); P3 <- c(3,0) ell <- EllipseFromThreeBoundaryPoints(P1, P2, P3) ell$includes(P1); ell$includes(P2); ell$includes(P3)
An arc is given by an ellipse (Ellipse object),
a starting angle and an ending angle. They are respectively named
ell, alpha1 and alpha2.
ellget or set the ellipse
alpha1get or set the starting angle
alpha2get or set the ending angle
degreesget or set the degrees field
new()
Create a new EllipticalArc object.
EllipticalArc$new(ell, alpha1, alpha2, degrees = TRUE)
ellthe ellipse
alpha1the starting angle
alpha2the ending angle
degreeslogical, whether alpha1 and alpha2 are
given in degrees
A new EllipticalArc object.
ell <- Ellipse$new(c(-4,0), 4, 2.5, 140) EllipticalArc$new(ell, 45, 90)
print()
Show instance of an EllipticalArc object.
EllipticalArc$print(...)
...ignored
startingPoint()
Starting point of the reference elliptical arc.
EllipticalArc$startingPoint()
endingPoint()
Ending point of the reference elliptical arc.
EllipticalArc$endingPoint()
isEqual()
Check whether the reference elliptical arc equals another elliptical arc.
EllipticalArc$isEqual(arc)
arcan EllipticalArc object
complementaryArc()
Complementary elliptical arc of the reference elliptical arc.
EllipticalArc$complementaryArc()
ell <- Ellipse$new(c(-4,0), 4, 2.5, 140)
arc <- EllipticalArc$new(ell, 30, 60)
plot(NULL, type = "n", asp = 1, xlim = c(-8,0), ylim = c(-3.2,3.2),
xlab = NA, ylab = NA)
draw(arc, lwd = 3, col = "red")
draw(arc$complementaryArc(), lwd = 3, col = "green")
path()
The reference elliptical arc as a path.
EllipticalArc$path(npoints = 100L)
npointsnumber of points of the path
A matrix with two columns x and y of length
npoints.
length()
The length of the elliptical arc.
EllipticalArc$length()
A number, the arc length.
clone()
The objects of this class are cloneable with this method.
EllipticalArc$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `EllipticalArc$new` ## ------------------------------------------------ ell <- Ellipse$new(c(-4,0), 4, 2.5, 140) EllipticalArc$new(ell, 45, 90) ## ------------------------------------------------ ## Method `EllipticalArc$complementaryArc` ## ------------------------------------------------ ell <- Ellipse$new(c(-4,0), 4, 2.5, 140) arc <- EllipticalArc$new(ell, 30, 60) plot(NULL, type = "n", asp = 1, xlim = c(-8,0), ylim = c(-3.2,3.2), xlab = NA, ylab = NA) draw(arc, lwd = 3, col = "red") draw(arc$complementaryArc(), lwd = 3, col = "green")## ------------------------------------------------ ## Method `EllipticalArc$new` ## ------------------------------------------------ ell <- Ellipse$new(c(-4,0), 4, 2.5, 140) EllipticalArc$new(ell, 45, 90) ## ------------------------------------------------ ## Method `EllipticalArc$complementaryArc` ## ------------------------------------------------ ell <- Ellipse$new(c(-4,0), 4, 2.5, 140) arc <- EllipticalArc$new(ell, 30, 60) plot(NULL, type = "n", asp = 1, xlim = c(-8,0), ylim = c(-3.2,3.2), xlab = NA, ylab = NA) draw(arc, lwd = 3, col = "red") draw(arc$complementaryArc(), lwd = 3, col = "green")
Fit an ellipse to a set of points.
fitEllipse(points)fitEllipse(points)
points |
numeric matrix with two columns, one point per row |
An Ellipse object representing the fitted ellipse. The
residual sum of squares is given in the RSS attribute.
library(PlaneGeometry) # We add some noise to 30 points on an ellipse: ell <- Ellipse$new(c(1, 1), 3, 2, 30) set.seed(666L) points <- ell$randomPoints(30, "on") + matrix(rnorm(30*2, sd = 0.2), ncol = 2) # Now we fit an ellipse to these points: ellFitted <- fitEllipse(points) # let's draw all this stuff: box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, border = "blue", lwd = 2) points(points, pch = 19) draw(ellFitted, border = "green", lwd = 2)library(PlaneGeometry) # We add some noise to 30 points on an ellipse: ell <- Ellipse$new(c(1, 1), 3, 2, 30) set.seed(666L) points <- ell$randomPoints(30, "on") + matrix(rnorm(30*2, sd = 0.2), ncol = 2) # Now we fit an ellipse to these points: ellFitted <- fitEllipse(points) # let's draw all this stuff: box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, border = "blue", lwd = 2) points(points, pch = 19) draw(ellFitted, border = "green", lwd = 2)
Return the ellipse equal to the highest pdf region of a bivariate Gaussian distribution with a given probability.
GaussianEllipse(mean, Sigma, p)GaussianEllipse(mean, Sigma, p)
mean |
numeric vector of length 2, the mean of the bivariate Gaussian distribution; this is the center of the ellipse |
Sigma |
covariance matrix of the bivariate Gaussian distribution |
p |
desired probability level, a number between 0 and 1 (strictly) |
An Ellipse object.
A homothety is given by a center and a scale factor.
centerget or set the center
scaleget or set the scale factor of the homothety
new()
Create a new Homothety object.
Homothety$new(center, scale)
centera point, the center of the homothety
scalea number, the scale factor of the homothety
A new Homothety object.
Homothety$new(c(1,1), 2)
print()
Show instance of a Homothety object.
Homothety$print(...)
...ignored
transform()
Transform a point or several points by the reference homothety.
Homothety$transform(M)
Ma point or a two-column matrix of points, one point per row
transformCircle()
Transform a circle by the reference homothety.
Homothety$transformCircle(circ)
circa Circle object
A Circle object.
getMatrix()
Augmented matrix of the homothety.
Homothety$getMatrix()
A 3x3 matrix.
H <- Homothety$new(c(1,1), 2) P <- c(1,5) H$transform(P) H$getMatrix() %*% c(P,1)
asAffine()
Convert the reference homothety to an Affine object.
Homothety$asAffine()
clone()
The objects of this class are cloneable with this method.
Homothety$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Homothety$new` ## ------------------------------------------------ Homothety$new(c(1,1), 2) ## ------------------------------------------------ ## Method `Homothety$getMatrix` ## ------------------------------------------------ H <- Homothety$new(c(1,1), 2) P <- c(1,5) H$transform(P) H$getMatrix() %*% c(P,1)## ------------------------------------------------ ## Method `Homothety$new` ## ------------------------------------------------ Homothety$new(c(1,1), 2) ## ------------------------------------------------ ## Method `Homothety$getMatrix` ## ------------------------------------------------ H <- Homothety$new(c(1,1), 2) P <- c(1,5) H$transform(P) H$getMatrix() %*% c(P,1)
A hyperbola is given by two intersecting asymptotes, named
L1 and L2, and a point on this hyperbola, named M.
L1get or set the asymptote L1
L2get or set the asymptote L2
Mget or set the point M
new()
Create a new Hyperbola object.
Hyperbola$new(L1, L2, M)
L1, L2two intersecting lines given as Line objects, the
asymptotes
Ma point on the hyperbola
A new Hyperbola object.
center()
Center of the hyperbola.
Hyperbola$center()
The center of the hyperbola, i.e. the point where the two asymptotes meet each other.
OAB()
Parametric equation
representing the hyperbola.
Hyperbola$OAB()
The point O and the two vectors A and B
in a list.
L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$OAB()
vertices()
Vertices of the hyperbola.
Hyperbola$vertices()
The two vertices V1 and V2 in a list.
abce()
The numbers a (semi-major axis, i.e. distance
from center to vertex),
b (semi-minor axis),
c (linear eccentricity)
and e (eccentricity)
associated to the hyperbola.
Hyperbola$abce()
The four numbers a, b, c and e
in a list.
foci()
Foci of the hyperbola.
Hyperbola$foci()
The two foci F1 and F2 in a list.
plot()
Plot hyperbola.
Hyperbola$plot(add = FALSE, ...)
addBoolean, whether to add this plot to the current plot
...named arguments passed to lines
Nothing, called for plotting.
L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) plot(hyperbola, lwd = 2) points(t(M), pch = 19, col = "blue") O <- hyperbola$center() points(t(O), pch = 19) draw(L1, col = "red") draw(L2, col = "red") vertices <- hyperbola$vertices() points(rbind(vertices$V1, vertices$V2), pch = 19) majorAxis <- Line$new(vertices$V1, vertices$V2) draw(majorAxis, lty = "dashed") foci <- hyperbola$foci() points(rbind(foci$F1, foci$F2), pch = 19, col = "green")
includes()
Whether a point belongs to the hyperbola.
Hyperbola$includes(P)
Pa point
A Boolean value.
L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$includes(M)
equation()
Implicit quadratic equation of the hyperbola Axxx2 + 2Axyxy + Ayyy2 + 2Bxx + 2Byy + C = 0
Hyperbola$equation()
The coefficients of the equation in a named list.
L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) eq <- hyperbola$equation() x <- M[1]; y <- M[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C) V1 <- hyperbola$vertices()$V1 x <- V1[1]; y <- V1[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C)
clone()
The objects of this class are cloneable with this method.
Hyperbola$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Hyperbola$OAB` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$OAB() ## ------------------------------------------------ ## Method `Hyperbola$plot` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) plot(hyperbola, lwd = 2) points(t(M), pch = 19, col = "blue") O <- hyperbola$center() points(t(O), pch = 19) draw(L1, col = "red") draw(L2, col = "red") vertices <- hyperbola$vertices() points(rbind(vertices$V1, vertices$V2), pch = 19) majorAxis <- Line$new(vertices$V1, vertices$V2) draw(majorAxis, lty = "dashed") foci <- hyperbola$foci() points(rbind(foci$F1, foci$F2), pch = 19, col = "green") ## ------------------------------------------------ ## Method `Hyperbola$includes` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$includes(M) ## ------------------------------------------------ ## Method `Hyperbola$equation` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) eq <- hyperbola$equation() x <- M[1]; y <- M[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C) V1 <- hyperbola$vertices()$V1 x <- V1[1]; y <- V1[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C)## ------------------------------------------------ ## Method `Hyperbola$OAB` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$OAB() ## ------------------------------------------------ ## Method `Hyperbola$plot` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) plot(hyperbola, lwd = 2) points(t(M), pch = 19, col = "blue") O <- hyperbola$center() points(t(O), pch = 19) draw(L1, col = "red") draw(L2, col = "red") vertices <- hyperbola$vertices() points(rbind(vertices$V1, vertices$V2), pch = 19) majorAxis <- Line$new(vertices$V1, vertices$V2) draw(majorAxis, lty = "dashed") foci <- hyperbola$foci() points(rbind(foci$F1, foci$F2), pch = 19, col = "green") ## ------------------------------------------------ ## Method `Hyperbola$includes` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) hyperbola$includes(M) ## ------------------------------------------------ ## Method `Hyperbola$equation` ## ------------------------------------------------ L1 <- LineFromInterceptAndSlope(0, 2) L2 <- LineFromInterceptAndSlope(-2, -0.5) M <- c(4, 3) hyperbola <- Hyperbola$new(L1, L2, M) eq <- hyperbola$equation() x <- M[1]; y <- M[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C) V1 <- hyperbola$vertices()$V1 x <- V1[1]; y <- V1[2] with(eq, Axx*x^2 + 2*Axy*x*y + Ayy*y^2 + 2*Bx*x + 2*By*y + C)
Create the Hyperbola object representing the hyperbola
with the given implicit equation.
HyperbolaFromEquation(eq)HyperbolaFromEquation(eq)
eq |
named vector or list of the six parameters |
A Hyperbola object.
Return the intersection of two circles.
intersectionCircleCircle(circ1, circ2, epsilon = sqrt(.Machine$double.eps))intersectionCircleCircle(circ1, circ2, epsilon = sqrt(.Machine$double.eps))
circ1, circ2
|
two |
epsilon |
a small positive number used for the numerical accuracy |
NULL if there is no intersection,
a point if the circles touch, a list of two points if the circles meet at
two points, a circle if the two circles are identical.
Return the intersection of a circle and a line.
intersectionCircleLine(circ, line, strict = FALSE)intersectionCircleLine(circ, line, strict = FALSE)
circ |
a |
line |
a |
strict |
logical, whether to take into account |
NULL if there is no intersection;
a point if the infinite line is tangent to the circle, or NULL
if strict=TRUE and the point is not on the line (segment or half-line);
a list of two points if the circle and the infinite line meet at
two points, when strict=FALSE; if strict=TRUE and the line is
a segment or a half-line, this can return NULL or a single point.
circ <- Circle$new(c(1,1), 2) line <- Line$new(c(2,-2), c(1,2), FALSE, FALSE) intersectionCircleLine(circ, line) intersectionCircleLine(circ, line, strict = TRUE)circ <- Circle$new(c(1,1), 2) line <- Line$new(c(2,-2), c(1,2), FALSE, FALSE) intersectionCircleLine(circ, line) intersectionCircleLine(circ, line, strict = TRUE)
Return the intersection of an ellipse and a line.
intersectionEllipseLine(ell, line, strict = FALSE)intersectionEllipseLine(ell, line, strict = FALSE)
ell |
an |
line |
a |
strict |
logical, whether to take into account |
NULL if there is no intersection;
a point if the infinite line is tangent to the ellipse, or NULL
if strict=TRUE and the point is not on the line (segment or half-line);
a list of two points if the ellipse and the infinite line meet at
two points, when strict=FALSE; if strict=TRUE and the line is
a segment or a half-line, this can return NULL or a single point.
ell <- Ellipse$new(c(1,1), 5, 1, 30) line <- Line$new(c(2,-2), c(0,4)) ( Is <- intersectionEllipseLine(ell, line) ) ell$includes(Is$I1); ell$includes(Is$I2)ell <- Ellipse$new(c(1,1), 5, 1, 30) line <- Line$new(c(2,-2), c(0,4)) ( Is <- intersectionEllipseLine(ell, line) ) ell$includes(Is$I1); ell$includes(Is$I2)
Return the intersection of two lines.
intersectionLineLine(line1, line2, strict = FALSE)intersectionLineLine(line1, line2, strict = FALSE)
line1, line2
|
two |
strict |
logical, whether to take into account the extensions of the
lines ( |
If strict = FALSE this returns either a point, or NULL
if the lines are parallel, or a bi-infinite line if the two lines coincide.
If strict = TRUE, this can also return a half-infinite line or
a segment.
An inversion is given by a pole (a point) and a power (a number, possibly negative, but not zero).
poleget or set the pole
powerget or set the power
new()
Create a new Inversion object.
Inversion$new(pole, power)
polethe pole
powerthe power
A new Inversion object.
print()
Show instance of an inversion object.
Inversion$print(...)
...ignored
Inversion$new(c(0,0), 2)
invert()
Inversion of a point.
Inversion$invert(M)
Ma point or Inf
A point or Inf, the image of M.
transform()
An alias of invert.
Inversion$transform(M)
Ma point or Inf
A point or Inf, the image of M.
invertCircle()
Inversion of a circle.
Inversion$invertCircle(circ)
circa Circle object
A Circle object or a Line object.
# A Pappus chain
# https://www.cut-the-knot.org/Curriculum/Geometry/InversionInArbelos.shtml
opar <- par(mar = c(0,0,0,0))
plot(0, 0, type = "n", asp = 1, xlim = c(0,6), ylim = c(-4,4),
xlab = NA, ylab = NA, axes = FALSE)
A <- c(0,0); B <- c(6,0)
ABsqr <- c(crossprod(A-B))
iota <- Inversion$new(A, ABsqr)
C <- iota$invert(c(8,0))
Sigma1 <- Circle$new((A+B)/2, sqrt(ABsqr)/2)
Sigma2 <- Circle$new((A+C)/2, sqrt(c(crossprod(A-C)))/2)
draw(Sigma1); draw(Sigma2)
circ0 <- Circle$new(c(7,0), 1)
iotacirc0 <- iota$invertCircle(circ0)
draw(iotacirc0)
for(i in 1:6){
circ <- circ0$translate(c(0,2*i))
iotacirc <- iota$invertCircle(circ)
draw(iotacirc)
circ <- circ0$translate(c(0,-2*i))
iotacirc <- iota$invertCircle(circ)
draw(iotacirc)
}
par(opar)
transformCircle()
An alias of invertCircle.
Inversion$transformCircle(circ)
circa Circle object
A Circle object or a Line object.
invertLine()
Inversion of a line.
Inversion$invertLine(line)
linea Line object
A Circle object or a Line object.
transformLine()
An alias of invertLine.
Inversion$transformLine(line)
linea Line object
A Circle object or a Line object.
invertGcircle()
Inversion of a generalized circle (i.e. a circle or a line).
Inversion$invertGcircle(gcircle)
gcirclea Circle object or a Line object
A Circle object or a Line object.
compose()
Compose the reference inversion with another inversion. The result is a Möbius transformation.
Inversion$compose(iota1, left = TRUE)
iota1an Inversion object
leftlogical, whether to compose at left or at right (i.e.
returns iota1 o iota0 or iota0 o iota1)
A Mobius object.
clone()
The objects of this class are cloneable with this method.
Inversion$clone(deep = FALSE)
deepWhether to make a deep clone.
inversionSwappingTwoCircles,
inversionFixingTwoCircles,
inversionFixingThreeCircles to create some inversions.
## ------------------------------------------------ ## Method `Inversion$print` ## ------------------------------------------------ Inversion$new(c(0,0), 2) ## ------------------------------------------------ ## Method `Inversion$invertCircle` ## ------------------------------------------------ # A Pappus chain # https://www.cut-the-knot.org/Curriculum/Geometry/InversionInArbelos.shtml opar <- par(mar = c(0,0,0,0)) plot(0, 0, type = "n", asp = 1, xlim = c(0,6), ylim = c(-4,4), xlab = NA, ylab = NA, axes = FALSE) A <- c(0,0); B <- c(6,0) ABsqr <- c(crossprod(A-B)) iota <- Inversion$new(A, ABsqr) C <- iota$invert(c(8,0)) Sigma1 <- Circle$new((A+B)/2, sqrt(ABsqr)/2) Sigma2 <- Circle$new((A+C)/2, sqrt(c(crossprod(A-C)))/2) draw(Sigma1); draw(Sigma2) circ0 <- Circle$new(c(7,0), 1) iotacirc0 <- iota$invertCircle(circ0) draw(iotacirc0) for(i in 1:6){ circ <- circ0$translate(c(0,2*i)) iotacirc <- iota$invertCircle(circ) draw(iotacirc) circ <- circ0$translate(c(0,-2*i)) iotacirc <- iota$invertCircle(circ) draw(iotacirc) } par(opar)## ------------------------------------------------ ## Method `Inversion$print` ## ------------------------------------------------ Inversion$new(c(0,0), 2) ## ------------------------------------------------ ## Method `Inversion$invertCircle` ## ------------------------------------------------ # A Pappus chain # https://www.cut-the-knot.org/Curriculum/Geometry/InversionInArbelos.shtml opar <- par(mar = c(0,0,0,0)) plot(0, 0, type = "n", asp = 1, xlim = c(0,6), ylim = c(-4,4), xlab = NA, ylab = NA, axes = FALSE) A <- c(0,0); B <- c(6,0) ABsqr <- c(crossprod(A-B)) iota <- Inversion$new(A, ABsqr) C <- iota$invert(c(8,0)) Sigma1 <- Circle$new((A+B)/2, sqrt(ABsqr)/2) Sigma2 <- Circle$new((A+C)/2, sqrt(c(crossprod(A-C)))/2) draw(Sigma1); draw(Sigma2) circ0 <- Circle$new(c(7,0), 1) iotacirc0 <- iota$invertCircle(circ0) draw(iotacirc0) for(i in 1:6){ circ <- circ0$translate(c(0,2*i)) iotacirc <- iota$invertCircle(circ) draw(iotacirc) circ <- circ0$translate(c(0,-2*i)) iotacirc <- iota$invertCircle(circ) draw(iotacirc) } par(opar)
Return the inversion which lets invariant three given circles.
inversionFixingThreeCircles(circ1, circ2, circ3)inversionFixingThreeCircles(circ1, circ2, circ3)
circ1, circ2, circ3
|
|
An Inversion object, which lets each of circ1,
circ2 and circ3 invariant.
Return the inversion which lets invariant two given circles.
inversionFixingTwoCircles(circ1, circ2)inversionFixingTwoCircles(circ1, circ2)
circ1, circ2
|
|
An Inversion object, which maps circ1 to circ2
and circ2 to circ2.
Return the inversion on a given circle.
inversionFromCircle(circ)inversionFromCircle(circ)
circ |
a |
An Inversion object
Return an inversion with a given pole which keeps a given circle unchanged.
inversionKeepingCircle(pole, circ)inversionKeepingCircle(pole, circ)
pole |
inversion pole, a point |
circ |
a |
An Inversion object.
circ <- Circle$new(c(4,3), 2) iota <- inversionKeepingCircle(c(1,2), circ) iota$transformCircle(circ)circ <- Circle$new(c(4,3), 2) iota <- inversionKeepingCircle(c(1,2), circ) iota$transformCircle(circ)
Return the inversion which swaps two given circles.
inversionSwappingTwoCircles(circ1, circ2, positive = TRUE)inversionSwappingTwoCircles(circ1, circ2, positive = TRUE)
circ1, circ2
|
|
positive |
logical, whether the sign of the desired inversion power must be positive or negative |
An Inversion object, which maps circ1 to circ2
and circ2 to circ1, except in the case when circ1
and circ2 are congruent and tangent: in this case a Reflection
object is returned (a reflection is an inversion on a line).
A line is given by two distinct points,
named A and B, and two logical values extendA
and extendB, indicating whether the line must be extended
beyond A and B respectively. Depending on extendA
and extendB, the line is an infinite line, a half-line, or a segment.
Aget or set the point A
Bget or set the point B
extendAget or set extendA
extendBget or set extendB
new()
Create a new Line object.
Line$new(A, B, extendA = TRUE, extendB = TRUE)
A, Bpoints
extendA, extendBlogical values
A new Line object.
l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) l l$A l$A <- c(0,0) l
print()
Show instance of a line object.
Line$print(...)
...ignored
Line$new(c(0,0), c(1,0), FALSE, TRUE)
length()
Segment length, returns the length of the segment joining the two points defining the line.
Line$length()
directionAndOffset()
Direction (angle between 0 and 2pi) and offset (positive number) of the reference line.
Line$directionAndOffset()
The equation of the line is cos(θ)x+sin(θ)y=d where θ is the direction and d is the offset.
isEqual()
Check whether the reference line equals a given line,
without taking into account extendA and extendB.
Line$isEqual(line)
linea Line object
TRUE or FALSE.
isParallel()
Check whether the reference line is parallel to a given line.
Line$isParallel(line)
linea Line object
TRUE or FALSE.
isPerpendicular()
Check whether the reference line is perpendicular to a given line.
Line$isPerpendicular(line)
linea Line object
TRUE or FALSE.
includes()
Whether a point belongs to the reference line.
Line$includes(M, strict = FALSE, checkCollinear = TRUE)
Mthe point for which we want to test whether it belongs to the line
strictlogical, whether to take into account extendA
and extendB
checkCollinearlogical, whether to check the collinearity of
A, B, M; set to FALSE only if you are sure
that M is on the line (AB) in case if you use
strict=TRUE
TRUE or FALSE.
A <- c(0,0); B <- c(1,2); M <- c(3,6) l <- Line$new(A, B, FALSE, FALSE) l$includes(M, strict = TRUE)
perpendicular()
Perpendicular line passing through a given point.
Line$perpendicular(M, extendH = FALSE, extendM = TRUE)
Mthe point through which the perpendicular passes.
extendHlogical, whether to extend the perpendicular line beyond the meeting point
extendMlogical, whether to extend the perpendicular line
beyond the point M
A Line object; its two points are the
meeting point and the point M.
parallel()
Parallel to the reference line passing through a given point.
Line$parallel(M)
Ma point
A Line object.
projection()
Orthogonal projection of a point to the reference line.
Line$projection(M)
Ma point
A point.
distance()
Distance from a point to the reference line.
Line$distance(M)
Ma point
A positive number.
reflection()
Reflection of a point with respect to the reference line.
Line$reflection(M)
Ma point
A point.
rotate()
Rotate the reference line.
Line$rotate(alpha, O, degrees = TRUE)
alphaangle of rotation
Ocenter of rotation
degreeslogical, whether alpha is given in degrees
A Line object.
translate()
Translate the reference line.
Line$translate(v)
vthe vector of translation
A Line object.
invert()
Invert the reference line.
Line$invert(inversion)
inversionan Inversion object
A Circle object or a Line object.
clone()
The objects of this class are cloneable with this method.
Line$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Line$new` ## ------------------------------------------------ l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) l l$A l$A <- c(0,0) l ## ------------------------------------------------ ## Method `Line$print` ## ------------------------------------------------ Line$new(c(0,0), c(1,0), FALSE, TRUE) ## ------------------------------------------------ ## Method `Line$includes` ## ------------------------------------------------ A <- c(0,0); B <- c(1,2); M <- c(3,6) l <- Line$new(A, B, FALSE, FALSE) l$includes(M, strict = TRUE)## ------------------------------------------------ ## Method `Line$new` ## ------------------------------------------------ l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) l l$A l$A <- c(0,0) l ## ------------------------------------------------ ## Method `Line$print` ## ------------------------------------------------ Line$new(c(0,0), c(1,0), FALSE, TRUE) ## ------------------------------------------------ ## Method `Line$includes` ## ------------------------------------------------ A <- c(0,0); B <- c(1,2); M <- c(3,6) l <- Line$new(A, B, FALSE, FALSE) l$includes(M, strict = TRUE)
Create a Line object representing the infinite line
with given equation .
LineFromEquation(a, b, c)LineFromEquation(a, b, c)
a, b, c
|
the parameters of the equation; |
A Line object.
Create a Line object representing the infinite line
with given intercept and given slope.
LineFromInterceptAndSlope(a, b)LineFromInterceptAndSlope(a, b)
a |
intercept |
b |
slope |
A Line object.
Minimum area ellipse containing a set of points.
LownerJohnEllipse(pts)LownerJohnEllipse(pts)
pts |
the points in a two-columns matrix (one point per row); at least three distinct points |
An Ellipse object.
pts <- cbind(rnorm(30, sd=2), rnorm(30)) ell <- LownerJohnEllipse(pts) box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, col = "seaShell") points(pts, pch = 19) all(apply(pts, 1, ell$contains)) # should be TRUEpts <- cbind(rnorm(30, sd=2), rnorm(30)) ell <- LownerJohnEllipse(pts) box <- ell$boundingbox() plot(NULL, asp = 1, xlim = box$x, ylim = box$y, xlab = NA, ylab = NA) draw(ell, col = "seaShell") points(pts, pch = 19) all(apply(pts, 1, ell$contains)) # should be TRUE
Computes the circle inscribed in a convex polygon with maximum area. This is the so-called Chebyshev circle.
maxAreaInscribedCircle(points, verbose = FALSE)maxAreaInscribedCircle(points, verbose = FALSE)
points |
the vertices of the polygon in a two-columns matrix; their order has no importance, since the procedure takes the convex hull of these points (and does not check the convexity) |
verbose |
argument passed to |
A Circle object. The status of the optimization problem
is given as an attribute of this circle. A warning is thrown if it is
not optimal.
library(PlaneGeometry) hexagon <- rbind( c(-1.7, -1), c(-1.4, 0.4), c(0.3, 1.3), c(1.7, 0.6), c(1.3, -0.3), c(-0.4, -1.8) ) opar <- par(mar = c(2, 2, 1, 1)) plot(NULL, xlim=c(-2, 2), ylim=c(-2, 2), xlab = NA, ylab = NA, asp = 1) points(hexagon, pch = 19) polygon(hexagon) circ <- maxAreaInscribedCircle(hexagon) draw(circ, col = "yellow2", border = "blue", lwd = 2) par(opar) # check optimization status: attr(circ, "status")library(PlaneGeometry) hexagon <- rbind( c(-1.7, -1), c(-1.4, 0.4), c(0.3, 1.3), c(1.7, 0.6), c(1.3, -0.3), c(-0.4, -1.8) ) opar <- par(mar = c(2, 2, 1, 1)) plot(NULL, xlim=c(-2, 2), ylim=c(-2, 2), xlab = NA, ylab = NA, asp = 1) points(hexagon, pch = 19) polygon(hexagon) circ <- maxAreaInscribedCircle(hexagon) draw(circ, col = "yellow2", border = "blue", lwd = 2) par(opar) # check optimization status: attr(circ, "status")
Computes the ellipse inscribed in a convex polygon with maximum area.
maxAreaInscribedEllipse(points, verbose = FALSE)maxAreaInscribedEllipse(points, verbose = FALSE)
points |
the vertices of the polygon in a two-columns matrix; their order has no importance, since the procedure takes the convex hull of these points (and does not check the convexity) |
verbose |
argument passed to |
An Ellipse object. The status of the optimization problem
is given as an attribute of this ellipse. A warning is thrown if it is
not optimal.
hexagon <- rbind( c(-1.7, -1), c(-1.4, 0.4), c(0.3, 1.3), c(1.7, 0.6), c(1.3, -0.3), c(-0.4, -1.8) ) opar <- par(mar = c(2, 2, 1, 1)) plot(NULL, xlim=c(-2, 2), ylim=c(-2, 2), xlab = NA, ylab = NA, asp = 1) points(hexagon, pch = 19) polygon(hexagon) ell <- maxAreaInscribedEllipse(hexagon) draw(ell, col = "yellow2", border = "blue", lwd = 2) par(opar) # check optimization status: attr(ell, "status")hexagon <- rbind( c(-1.7, -1), c(-1.4, 0.4), c(0.3, 1.3), c(1.7, 0.6), c(1.3, -0.3), c(-0.4, -1.8) ) opar <- par(mar = c(2, 2, 1, 1)) plot(NULL, xlim=c(-2, 2), ylim=c(-2, 2), xlab = NA, ylab = NA, asp = 1) points(hexagon, pch = 19) polygon(hexagon) ell <- maxAreaInscribedEllipse(hexagon) draw(ell, col = "yellow2", border = "blue", lwd = 2) par(opar) # check optimization status: attr(ell, "status")
Return the mid-circle(s) of two circles.
midCircles(circ1, circ2)midCircles(circ1, circ2)
circ1, circ2
|
|
A mid-circle of two circles is a generalized circle (i.e. a circle or a line) such that the inversion on this circle swaps the two circles. The case of a line appears only when the two circles have equal radii.
A Circle object, or a Line object, or a list of two
such objects.
circ1 <- Circle$new(c(5,4),2) circ2 <- Circle$new(c(6,4),1) midcircle <- midCircles(circ1, circ2) inversionFromCircle(midcircle) inversionSwappingTwoCircles(circ1, circ2)circ1 <- Circle$new(c(5,4),2) circ2 <- Circle$new(c(6,4),1) midcircle <- midCircles(circ1, circ2) inversionFromCircle(midcircle) inversionSwappingTwoCircles(circ1, circ2)
A Möbius transformation is given by a matrix of complex numbers with non-null determinant.
aget or set a
bget or set b
cget or set c
dget or set d
new()
Create a new Mobius object.
Mobius$new(M)
Mthe matrix corresponding to the Möbius transformation
A new Mobius object.
print()
Show instance of a Mobius object.
Mobius$print(...)
...ignored
Mobius$new(rbind(c(1+1i,2),c(0,3-2i)))
getM()
Get the matrix corresponding to the Möbius transformation.
Mobius$getM()
compose()
Compose the reference Möbius transformation with another Möbius transformation
Mobius$compose(M1, left = TRUE)
M1a Mobius object
leftlogical, whether to compose at left or at right (i.e.
returns M1 o M0 or M0 o M1)
A Mobius object.
inverse()
Inverse of the reference Möbius transformation.
Mobius$inverse()
A Mobius object.
power()
Power of the reference Möbius transformation.
Mobius$power(k)
kan integer, possibly negative
The Möbius transformation M^k,
where M is the reference Möbius transformation.
gpower()
Generalized power of the reference Möbius transformation.
Mobius$gpower(k)
ka real number, possibly negative
A Mobius object, the generalized k-th power of
the reference Möbius transformation.
M <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mroot <- M$gpower(1/2) Mroot$compose(Mroot) # should be M
transform()
Transformation of a point by the reference Möbius transformation.
Mobius$transform(M)
Ma point or Inf
A point or Inf, the image of M.
Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mob$transform(c(1,1)) Mob$transform(Inf)
fixedPoints()
Returns the fixed points of the reference Möbius transformation.
Mobius$fixedPoints()
One point, or a list of two points, or a message in the case when the transformation is the identity map.
transformCircle()
Transformation of a circle by the reference Möbius transformation.
Mobius$transformCircle(circ)
circa Circle object
A Circle object or a Line object.
transformLine()
Transformation of a line by the reference Möbius transformation.
Mobius$transformLine(line)
linea Line object
A Circle object or a Line object.
transformGcircle()
Transformation of a generalized circle (i.e. a circle or a line) by the reference Möbius transformation.
Mobius$transformGcircle(gcirc)
gcirca Circle object or a Line object
A Circle object or a Line object.
clone()
The objects of this class are cloneable with this method.
Mobius$clone(deep = FALSE)
deepWhether to make a deep clone.
MobiusMappingThreePoints to create a Möbius
transformation, and also the compose method of the
Inversion R6 class.
## ------------------------------------------------ ## Method `Mobius$print` ## ------------------------------------------------ Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) ## ------------------------------------------------ ## Method `Mobius$gpower` ## ------------------------------------------------ M <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mroot <- M$gpower(1/2) Mroot$compose(Mroot) # should be M ## ------------------------------------------------ ## Method `Mobius$transform` ## ------------------------------------------------ Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mob$transform(c(1,1)) Mob$transform(Inf)## ------------------------------------------------ ## Method `Mobius$print` ## ------------------------------------------------ Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) ## ------------------------------------------------ ## Method `Mobius$gpower` ## ------------------------------------------------ M <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mroot <- M$gpower(1/2) Mroot$compose(Mroot) # should be M ## ------------------------------------------------ ## Method `Mobius$transform` ## ------------------------------------------------ Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mob$transform(c(1,1)) Mob$transform(Inf)
Returns a Möbius transformation mapping a given circle to another given circle.
MobiusMappingCircle(circ1, circ2)MobiusMappingCircle(circ1, circ2)
circ1, circ2
|
|
A Möbius transformation which maps circ1 to circ2.
library(PlaneGeometry) C1 <- Circle$new(c(0, 0), 1) C2 <- Circle$new(c(1, 2), 3) M <- MobiusMappingCircle(C1, C2) C3 <- M$transformCircle(C1) C3$isEqual(C2)library(PlaneGeometry) C1 <- Circle$new(c(0, 0), 1) C2 <- Circle$new(c(1, 2), 3) M <- MobiusMappingCircle(C1, C2) C3 <- M$transformCircle(C1) C3$isEqual(C2)
Return a Möbius transformation which sends
P1 to Q1, P2 to Q2 and P3 to Q3.
MobiusMappingThreePoints(P1, P2, P3, Q1, Q2, Q3)MobiusMappingThreePoints(P1, P2, P3, Q1, Q2, Q3)
P1, P2, P3
|
three distinct points, |
Q1, Q2, Q3
|
three distinct points, |
A Mobius object.
Return a Möbius transformation which sends
A to B and B to A.
MobiusSwappingTwoPoints(A, B)MobiusSwappingTwoPoints(A, B)
A, B
|
two distinct points, |
A Mobius object.
A projection on a line D parallel to another line
Delta is given by the line of projection (D)
and the directrix line (Delta).
Dget or set the projection line
Deltaget or set the directrix line
new()
Create a new Projection object.
Projection$new(D, Delta)
D, Deltatwo Line objects such that the two lines meet
(not parallel); or Delta = NULL for orthogonal projection onto
D
A new Projection object.
D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) Projection$new(D, Delta)
print()
Show instance of a projection object.
Projection$print(...)
...ignored
project()
Project a point.
Projection$project(M)
Ma point
D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) P <- Projection$new(D, Delta) M <- c(1,3) Mprime <- P$project(M) D$includes(Mprime) # should be TRUE Delta$isParallel(Line$new(M, Mprime)) # should be TRUE
transform()
An alias of project.
Projection$transform(M)
Ma point
getMatrix()
Augmented matrix of the projection.
Projection$getMatrix()
A 3x3 matrix.
P <- Projection$new(Line$new(c(2,2), c(4,5)), Line$new(c(0,0), c(1,1))) M <- c(1,5) P$project(M) P$getMatrix() %*% c(M,1)
asAffine()
Convert the reference projection to an Affine
object.
Projection$asAffine()
clone()
The objects of this class are cloneable with this method.
Projection$clone(deep = FALSE)
deepWhether to make a deep clone.
For an orthogonal projection, you can use the projection
method of the Line R6 class.
## ------------------------------------------------ ## Method `Projection$new` ## ------------------------------------------------ D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) Projection$new(D, Delta) ## ------------------------------------------------ ## Method `Projection$project` ## ------------------------------------------------ D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) P <- Projection$new(D, Delta) M <- c(1,3) Mprime <- P$project(M) D$includes(Mprime) # should be TRUE Delta$isParallel(Line$new(M, Mprime)) # should be TRUE ## ------------------------------------------------ ## Method `Projection$getMatrix` ## ------------------------------------------------ P <- Projection$new(Line$new(c(2,2), c(4,5)), Line$new(c(0,0), c(1,1))) M <- c(1,5) P$project(M) P$getMatrix() %*% c(M,1)## ------------------------------------------------ ## Method `Projection$new` ## ------------------------------------------------ D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) Projection$new(D, Delta) ## ------------------------------------------------ ## Method `Projection$project` ## ------------------------------------------------ D <- Line$new(c(1,1), c(5,5)) Delta <- Line$new(c(0,0), c(3,4)) P <- Projection$new(D, Delta) M <- c(1,3) Mprime <- P$project(M) D$includes(Mprime) # should be TRUE Delta$isParallel(Line$new(M, Mprime)) # should be TRUE ## ------------------------------------------------ ## Method `Projection$getMatrix` ## ------------------------------------------------ P <- Projection$new(Line$new(c(2,2), c(4,5)), Line$new(c(0,0), c(1,1))) M <- c(1,5) P$project(M) P$getMatrix() %*% c(M,1)
Returns the radical center of three circles.
radicalCenter(circ1, circ2, circ3)radicalCenter(circ1, circ2, circ3)
circ1, circ2, circ3
|
|
A point.
A reflection is given by a line.
lineget or set the line of the reflection
new()
Create a new Reflection object.
Reflection$new(line)
linea Line object
A new Reflection object.
l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) Reflection$new(l)
print()
Show instance of a reflection object.
Reflection$print(...)
...ignored
reflect()
Reflect a point.
Reflection$reflect(M)
Ma point, Inf allowed
transform()
An alias of reflect.
Reflection$transform(M)
Ma point, Inf allowed
reflectCircle()
Reflect a circle.
Reflection$reflectCircle(circ)
circa Circle object
A Circle object.
transformCircle()
An alias of reflectCircle.
Reflection$transformCircle(circ)
circa Circle object
A Circle object.
reflectLine()
Reflect a line.
Reflection$reflectLine(line)
linea Line object
A Line object.
transformLine()
An alias of reflectLine.
Reflection$transformLine(line)
linea Line object
A Line object.
getMatrix()
Augmented matrix of the reflection.
Reflection$getMatrix()
A 3x3 matrix.
R <- Reflection$new(Line$new(c(2,2), c(4,5))) P <- c(1,5) R$reflect(P) R$getMatrix() %*% c(P,1)
asAffine()
Convert the reference reflection to an Affine object.
Reflection$asAffine()
clone()
The objects of this class are cloneable with this method.
Reflection$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Reflection$new` ## ------------------------------------------------ l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) Reflection$new(l) ## ------------------------------------------------ ## Method `Reflection$getMatrix` ## ------------------------------------------------ R <- Reflection$new(Line$new(c(2,2), c(4,5))) P <- c(1,5) R$reflect(P) R$getMatrix() %*% c(P,1)## ------------------------------------------------ ## Method `Reflection$new` ## ------------------------------------------------ l <- Line$new(c(1,1), c(1.5,1.5), FALSE, TRUE) Reflection$new(l) ## ------------------------------------------------ ## Method `Reflection$getMatrix` ## ------------------------------------------------ R <- Reflection$new(Line$new(c(2,2), c(4,5))) P <- c(1,5) R$reflect(P) R$getMatrix() %*% c(P,1)
A rotation is given by an angle (theta) and a center.
thetaget or set the angle of the rotation
centerget or set the center
degreesget or set the degrees field
new()
Create a new Rotation object.
Rotation$new(theta, center, degrees = TRUE)
thetaa number, the angle of the rotation
centera point, the center of the rotation
degreeslogical, whether theta is given in degrees
A new Rotation object.
Rotation$new(60, c(1,1))
print()
Show instance of a Rotation object.
Rotation$print(...)
...ignored
rotate()
Rotate a point or several points.
Rotation$rotate(M)
Ma point or a two-column matrix of points, one point per row
transform()
An alias of rotate.
Rotation$transform(M)
Ma point or a two-column matrix of points, one point per row
rotateCircle()
Rotate a circle.
Rotation$rotateCircle(circ)
circa Circle object
A Circle object.
transformCircle()
An alias of rotateCircle.
Rotation$transformCircle(circ)
circa Circle object
A Circle object.
rotateEllipse()
Rotate an ellipse.
Rotation$rotateEllipse(ell)
ellan Ellipse object
An Ellipse object.
transformEllipse()
An alias of rotateEllipse.
Rotation$transformEllipse(ell)
ellan Ellipse object
An Ellipse object.
rotateLine()
Rotate a line.
Rotation$rotateLine(line)
linea Line object
A Line object.
transformLine()
An alias of rotateLine.
Rotation$transformLine(line)
linea Line object
A Line object.
getMatrix()
Augmented matrix of the rotation.
Rotation$getMatrix()
A 3x3 matrix.
R <- Rotation$new(60, c(1,1)) P <- c(1,5) R$rotate(P) R$getMatrix() %*% c(P,1)
asAffine()
Convert the reference rotation to an Affine object.
Rotation$asAffine()
clone()
The objects of this class are cloneable with this method.
Rotation$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `Rotation$new` ## ------------------------------------------------ Rotation$new(60, c(1,1)) ## ------------------------------------------------ ## Method `Rotation$getMatrix` ## ------------------------------------------------ R <- Rotation$new(60, c(1,1)) P <- c(1,5) R$rotate(P) R$getMatrix() %*% c(P,1)## ------------------------------------------------ ## Method `Rotation$new` ## ------------------------------------------------ Rotation$new(60, c(1,1)) ## ------------------------------------------------ ## Method `Rotation$getMatrix` ## ------------------------------------------------ R <- Rotation$new(60, c(1,1)) P <- c(1,5) R$rotate(P) R$getMatrix() %*% c(P,1)
A (non-uniform) scaling is given by a center, a direction vector, and a scale factor.
centerget or set the center
directionget or set the direction
scaleget or set the scale factor
new()
Create a new Scaling object.
Scaling$new(center, direction, scale)
centera point, the center of the scaling
directiona vector, the direction of the scaling
scalea number, the scale factor
A new Scaling object.
Scaling$new(c(1,1), c(1,3), 2)
print()
Show instance of a Scaling object.
Scaling$print(...)
...ignored
transform()
Transform a point or several points by the reference scaling.
Scaling$transform(M)
Ma point or a two-column matrix of points, one point per row
getMatrix()
Augmented matrix of the scaling.
Scaling$getMatrix()
A 3x3 matrix.
S <- Scaling$new(c(1,1), c(2,3), 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)
asAffine()
Convert the reference scaling to an Affine object.
Scaling$asAffine()
scaleCircle()
Scale a circle. The result is an ellipse.
Scaling$scaleCircle(circ)
circa Circle object
An Ellipse object.
clone()
The objects of this class are cloneable with this method.
Scaling$clone(deep = FALSE)
deepWhether to make a deep clone.
R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press, 2009.
Q <- c(1,1); w <- c(1,3); s <- 2 S <- Scaling$new(Q, w, s) # the center is mapped to itself: S$transform(Q) # any vector \code{u} parallel to the direction vector is mapped to \code{s*u}: u <- 3*w all.equal(s*u, S$transform(u) - S$transform(c(0,0))) # any vector perpendicular to the direction vector is mapped to itself wt <- 3*c(-w[2], w[1]) all.equal(wt, S$transform(wt) - S$transform(c(0,0))) ## ------------------------------------------------ ## Method `Scaling$new` ## ------------------------------------------------ Scaling$new(c(1,1), c(1,3), 2) ## ------------------------------------------------ ## Method `Scaling$getMatrix` ## ------------------------------------------------ S <- Scaling$new(c(1,1), c(2,3), 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)Q <- c(1,1); w <- c(1,3); s <- 2 S <- Scaling$new(Q, w, s) # the center is mapped to itself: S$transform(Q) # any vector \code{u} parallel to the direction vector is mapped to \code{s*u}: u <- 3*w all.equal(s*u, S$transform(u) - S$transform(c(0,0))) # any vector perpendicular to the direction vector is mapped to itself wt <- 3*c(-w[2], w[1]) all.equal(wt, S$transform(wt) - S$transform(c(0,0))) ## ------------------------------------------------ ## Method `Scaling$new` ## ------------------------------------------------ Scaling$new(c(1,1), c(1,3), 2) ## ------------------------------------------------ ## Method `Scaling$getMatrix` ## ------------------------------------------------ S <- Scaling$new(c(1,1), c(2,3), 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)
An axis-scaling is given by a center, and two scale factors
sx and sy, one for the x-axis and one for the y-axis.
centerget or set the center
sxget or set the scale factor of the x-axis
syget or set the scale factor of the y-ayis
new()
Create a new ScalingXY object.
ScalingXY$new(center, sx, sy)
centera point, the center of the scaling
sxa number, the scale factor of the x-axis
sya number, the scale factor of the y-axis
A new ScalingXY object.
ScalingXY$new(c(1,1), 4, 2)
print()
Show instance of a ScalingXY object.
ScalingXY$print(...)
...ignored
transform()
Transform a point or several points by the reference axis-scaling.
ScalingXY$transform(M)
Ma point or a two-column matrix of points, one point per row
A point or a two-column matrix of points.
getMatrix()
Augmented matrix of the axis-scaling.
ScalingXY$getMatrix()
A 3x3 matrix.
S <- ScalingXY$new(c(1,1), 4, 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)
asAffine()
Convert the reference axis-scaling to an Affine
object.
ScalingXY$asAffine()
clone()
The objects of this class are cloneable with this method.
ScalingXY$clone(deep = FALSE)
deepWhether to make a deep clone.
## ------------------------------------------------ ## Method `ScalingXY$new` ## ------------------------------------------------ ScalingXY$new(c(1,1), 4, 2) ## ------------------------------------------------ ## Method `ScalingXY$getMatrix` ## ------------------------------------------------ S <- ScalingXY$new(c(1,1), 4, 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)## ------------------------------------------------ ## Method `ScalingXY$new` ## ------------------------------------------------ ScalingXY$new(c(1,1), 4, 2) ## ------------------------------------------------ ## Method `ScalingXY$getMatrix` ## ------------------------------------------------ S <- ScalingXY$new(c(1,1), 4, 2) P <- c(1,5) S$transform(P) S$getMatrix() %*% c(P,1)
A shear is given by a vertex, two perpendicular vectors, and an angle.
vertexget or set the vertex
vectorget or set the first vector
ratioget or set the ratio between the length of vector
and the length of the second vector, perpendicular to the first one
angleget or set the angle
degreesget or set the degrees field
new()
Create a new Shear object.
Shear$new(vertex, vector, ratio, angle, degrees = TRUE)
vertexa point
vectora vector
ratioa positive number, the ratio between the length of vector
and the length of the second vector, perpendicular to the first one
anglean angle strictly between -90 degrees and 90 degrees
degreeslogical, whether angle is given in degrees
A new Shear object.
Shear$new(c(1,1), c(1,3), 0.5, 30)
print()
Show instance of a Shear object.
Shear$print(...)
...ignored
transform()
Transform a point or several points by the reference shear.
Shear$transform(M)
Ma point or a two-column matrix of points, one point per row
getMatrix()
Augmented matrix of the shear.
Shear$getMatrix()
A 3x3 matrix.
S <- Shear$new(c(1,1), c(1,3), 0.5, 30) S$getMatrix()
asAffine()
Convert the reference shear to an Affine object.
Shear$asAffine()
Shear$new(c(0,0), c(1,0), 1, atan(30), FALSE)$asAffine()
clone()
The objects of this class are cloneable with this method.
Shear$clone(deep = FALSE)
deepWhether to make a deep clone.
R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press, 2009.
P <- c(0,0); w <- c(1,0); ratio <- 1; angle <- 45 shear <- Shear$new(P, w, ratio, angle) wt <- ratio * c(-w[2], w[1]) Q <- P + w; R <- Q + wt; S <- P + wt A <- shear$transform(P) B <- shear$transform(Q) C <- shear$transform(R) D <- shear$transform(S) plot(0, 0, type = "n", asp = 1, xlim = c(0,1), ylim = c(0,2)) lines(rbind(P,Q,R,S,P), lwd = 2) # unit square lines(rbind(A,B,C,D,A), lwd = 2, col = "blue") # image by the shear ## ------------------------------------------------ ## Method `Shear$new` ## ------------------------------------------------ Shear$new(c(1,1), c(1,3), 0.5, 30) ## ------------------------------------------------ ## Method `Shear$getMatrix` ## ------------------------------------------------ S <- Shear$new(c(1,1), c(1,3), 0.5, 30) S$getMatrix() ## ------------------------------------------------ ## Method `Shear$asAffine` ## ------------------------------------------------ Shear$new(c(0,0), c(1,0), 1, atan(30), FALSE)$asAffine()P <- c(0,0); w <- c(1,0); ratio <- 1; angle <- 45 shear <- Shear$new(P, w, ratio, angle) wt <- ratio * c(-w[2], w[1]) Q <- P + w; R <- Q + wt; S <- P + wt A <- shear$transform(P) B <- shear$transform(Q) C <- shear$transform(R) D <- shear$transform(S) plot(0, 0, type = "n", asp = 1, xlim = c(0,1), ylim = c(0,2)) lines(rbind(P,Q,R,S,P), lwd = 2) # unit square lines(rbind(A,B,C,D,A), lwd = 2, col = "blue") # image by the shear ## ------------------------------------------------ ## Method `Shear$new` ## ------------------------------------------------ Shear$new(c(1,1), c(1,3), 0.5, 30) ## ------------------------------------------------ ## Method `Shear$getMatrix` ## ------------------------------------------------ S <- Shear$new(c(1,1), c(1,3), 0.5, 30) S$getMatrix() ## ------------------------------------------------ ## Method `Shear$asAffine` ## ------------------------------------------------ Shear$new(c(0,0), c(1,0), 1, atan(30), FALSE)$asAffine()
Inner Soddy circles associated to three circles.
soddyCircle(circ1, circ2, circ3)soddyCircle(circ1, circ2, circ3)
circ1, circ2, circ3
|
distinct circles |
A Circle object.
Return a Steiner chain of circles.
SteinerChain(c0, n, phi, shift, ellipse = FALSE)SteinerChain(c0, n, phi, shift, ellipse = FALSE)
c0 |
exterior circle, a |
n |
number of circles, not including the inner circle; at least |
phi |
|
shift |
any number; it produces a kind of rotation around the inner
circle; values between |
ellipse |
logical; the centers of the circles of the Steiner chain lie
on an ellipse, and this ellipse is returned as an attribute if you set this
argument to |
A list of n+1 Circle objects. The inner circle is stored at the
last position.
c0 <- Circle$new(c(1,1), 3) chain <- SteinerChain(c0, 5, 0.3, 0.5, ellipse = TRUE) plot(0, 0, type = "n", asp = 1, xlim = c(-4,4), ylim = c(-4,4)) invisible(lapply(chain, draw, lwd = 2, border = "blue")) draw(c0, lwd = 2) draw(attr(chain, "ellipse"), lwd = 2, border = "red")c0 <- Circle$new(c(1,1), 3) chain <- SteinerChain(c0, 5, 0.3, 0.5, ellipse = TRUE) plot(0, 0, type = "n", asp = 1, xlim = c(-4,4), ylim = c(-4,4)) invisible(lapply(chain, draw, lwd = 2, border = "blue")) draw(c0, lwd = 2) draw(attr(chain, "ellipse"), lwd = 2, border = "red")
A translation is given by a vector v.
vget or set the vector of translation
new()
Create a new Translation object.
Translation$new(v)
va numeric vector of length two, the vector of translation
A new Translation object.
print()
Show instance of a translation object.
Translation$print(...)
...ignored
project()
Transform a point or several points by the reference translation.
Translation$project(M)
Ma point or a two-column matrix of points, one point per row
transform()
An alias of translate.
Translation$transform(M)
Ma point or a two-column matrix of points, one point per row
translateLine()
Translate a line.
Translation$translateLine(line)
linea Line object
A Line object.
transformLine()
An alias of translateLine.
Translation$transformLine(line)
linea Line object
A Line object.
translateEllipse()
Translate a circle or an ellipse.
Translation$translateEllipse(ell)
ellan Ellipse object or a Circle object
An Ellipse object or a Circle object.
transformEllipse()
An alias of translateEllipse.
Translation$transformEllipse(ell)
ellan Ellipse object or a Circle object
An Ellipse object or a Circle object.
getMatrix()
Augmented matrix of the translation.
Translation$getMatrix()
A 3x3 matrix.
asAffine()
Convert the reference translation to an Affine
object.
Translation$asAffine()
clone()
The objects of this class are cloneable with this method.
Translation$clone(deep = FALSE)
deepWhether to make a deep clone.
A triangle has three vertices. They are named A, B, C.
Aget or set the vertex A
Bget or set the vertex B
Cget or set the vertex C
new()
Create a new Triangle object.
Triangle$new(A, B, C)
A, B, Cvertices
A new Triangle object.
t <- Triangle$new(c(0,0), c(1,0), c(1,1)) t t$C t$C <- c(2,2) t
print()
Show instance of a triangle object
Triangle$print(...)
...ignored
Triangle$new(c(0,0), c(1,0), c(1,1))
flatness()
Flatness of the triangle.
Triangle$flatness()
A number between 0 and 1. A triangle is flat when its flatness is 1.
a()
Length of the side BC.
Triangle$a()
b()
Length of the side AC.
Triangle$b()
c()
Length of the side AB.
Triangle$c()
edges()
The lengths of the sides of the triangle.
Triangle$edges()
A named numeric vector.
perimeter()
Perimeter of the triangle.
Triangle$perimeter()
The perimeter of the triangle.
orientation()
Determine the orientation of the triangle.
Triangle$orientation()
An integer: 1 for counterclockwise, -1 for clockwise, 0 for collinear.
contains()
Determine whether a point lies inside the reference triangle.
Triangle$contains(M)
Ma point
isAcute()
Determines whether the reference triangle is acute.
Triangle$isAcute()
'TRUE' if the triangle is acute (or right), 'FALSE' otherwise.
angleA()
Angle at the vertex A.
Triangle$angleA()
The angle at the vertex A in radians.
angleB()
Angle at the vertex B.
Triangle$angleB()
The angle at the vertex B in radians.
angleC()
Angle at the vertex C.
Triangle$angleC()
The angle at the vertex C in radians.
angles()
The three angles of the triangle.
Triangle$angles()
A named vector containing the values of the angles in radians.
X175()
Isoperimetric point, also known as the X(175) triangle center; this is the center of the outer Soddy circle.
Triangle$X175()
VeldkampIsoperimetricPoint()
Isoperimetric point in the sense of Veldkamp.
Triangle$VeldkampIsoperimetricPoint()
The isoperimetric point in the sense of Veldkamp, if it exists. Otherwise, returns 'NULL'.
centroid()
Centroid.
Triangle$centroid()
orthocenter()
Orthocenter.
Triangle$orthocenter()
area()
Area of the triangle.
Triangle$area()
incircle()
Incircle of the triangle.
Triangle$incircle()
A Circle object.
inradius()
Inradius of the reference triangle.
Triangle$inradius()
incenter()
Incenter of the reference triangle.
Triangle$incenter()
excircles()
Excircles of the triangle.
Triangle$excircles()
A list with the three excircles, Circle objects.
excentralTriangle()
Excentral triangle of the reference triangle.
Triangle$excentralTriangle()
A Triangle object.
BevanPoint()
Bevan point. This is the circumcenter of the excentral triangle.
Triangle$BevanPoint()
medialTriangle()
Medial triangle. Its vertices are the mid-points of the sides of the reference triangle.
Triangle$medialTriangle()
orthicTriangle()
Orthic triangle. Its vertices are the feet of the altitudes of the reference triangle.
Triangle$orthicTriangle()
incentralTriangle()
Incentral triangle.
Triangle$incentralTriangle()
It is the triangle whose vertices are the intersections of the reference triangle's angle bisectors with the respective opposite sides.
A Triangle object.
NagelTriangle()
Nagel triangle (or extouch triangle) of the reference triangle.
Triangle$NagelTriangle(NagelPoint = FALSE)
NagelPointlogical, whether to return the Nagel point as attribute
A Triangle object.
t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6))
lineAB <- Line$new(t$A, t$B)
lineAC <- Line$new(t$A, t$C)
lineBC <- Line$new(t$B, t$C)
NagelTriangle <- t$NagelTriangle(NagelPoint = TRUE)
NagelPoint <- attr(NagelTriangle, "Nagel point")
excircles <- t$excircles()
opar <- par(mar = c(0,0,0,0))
plot(0, 0, type="n", asp = 1, xlim = c(-1,5), ylim = c(-3,3),
xlab = NA, ylab = NA, axes = FALSE)
draw(t, lwd = 2)
draw(lineAB); draw(lineAC); draw(lineBC)
draw(excircles$A, border = "orange")
draw(excircles$B, border = "orange")
draw(excircles$C, border = "orange")
draw(NagelTriangle, lwd = 2, col = "red")
draw(Line$new(t$A, NagelTriangle$A, FALSE, FALSE), col = "blue")
draw(Line$new(t$B, NagelTriangle$B, FALSE, FALSE), col = "blue")
draw(Line$new(t$C, NagelTriangle$C, FALSE, FALSE), col = "blue")
points(rbind(NagelPoint), pch = 19)
par(opar)
NagelPoint()
Nagel point of the triangle.
Triangle$NagelPoint()
GergonneTriangle()
Gergonne triangle of the reference triangle.
Triangle$GergonneTriangle(GergonnePoint = FALSE)
GergonnePointlogical, whether to return the Gergonne point as an attribute
The Gergonne triangle is also known as the intouch triangle or the contact triangle. This is the triangle made of the three tangency points of the incircle.
A Triangle object.
GergonnePoint()
Gergonne point of the reference triangle.
Triangle$GergonnePoint()
tangentialTriangle()
Tangential triangle of the reference triangle. This is the triangle formed by the lines tangent to the circumcircle of the reference triangle at its vertices. It does not exist for a right triangle.
Triangle$tangentialTriangle()
A Triangle object.
symmedialTriangle()
Symmedial triangle of the reference triangle.
Triangle$symmedialTriangle()
A Triangle object.
t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6))
symt <- t$symmedialTriangle()
symmedianA <- Line$new(t$A, symt$A, FALSE, FALSE)
symmedianB <- Line$new(t$B, symt$B, FALSE, FALSE)
symmedianC <- Line$new(t$C, symt$C, FALSE, FALSE)
K <- t$symmedianPoint()
opar <- par(mar = c(0,0,0,0))
plot(NULL, asp = 1, xlim = c(-1,5), ylim = c(-3,3),
xlab = NA, ylab = NA, axes = FALSE)
draw(t, lwd = 2)
draw(symmedianA, lwd = 2, col = "blue")
draw(symmedianB, lwd = 2, col = "blue")
draw(symmedianC, lwd = 2, col = "blue")
points(rbind(K), pch = 19, col = "red")
par(opar)
symmedianPoint()
Symmedian point of the reference triangle.
Triangle$symmedianPoint()
A point.
circumcircle()
Circumcircle of the reference triangle.
Triangle$circumcircle()
A Circle object.
circumcenter()
Circumcenter of the reference triangle.
Triangle$circumcenter()
circumradius()
Circumradius of the reference triangle.
Triangle$circumradius()
BrocardCircle()
The Brocard circle of the reference triangle (also known as the seven-point circle).
Triangle$BrocardCircle()
A Circle object.
BrocardPoints()
Brocard points of the reference triangle.
Triangle$BrocardPoints()
A list of two points, the first Brocard point and the second Brocard point.
LemoineCircleI()
The first Lemoine circle of the reference triangle.
Triangle$LemoineCircleI()
A Circle object.
LemoineCircleII()
The second Lemoine circle of the reference triangle (also known as the cosine circle)
Triangle$LemoineCircleII()
A Circle object.
LemoineTriangle()
The Lemoine triangle of the reference triangle.
Triangle$LemoineTriangle()
A Triangle object.
LemoineCircleIII()
The third Lemoine circle of the reference triangle.
Triangle$LemoineCircleIII()
A Circle object.
ParryCircle()
Parry circle of the reference triangle.
Triangle$ParryCircle()
A Circle object.
outerSoddyCircle()
Soddy outer circle of the reference triangle.
Triangle$outerSoddyCircle()
A Circle object.
pedalTriangle()
Pedal triangle of a point with respect to the reference
triangle. The pedal triangle of a point P is the triangle whose
vertices are the feet of the perpendiculars from P to the sides
of the reference triangle.
Triangle$pedalTriangle(P)
Pa point
A Triangle object.
CevianTriangle()
Cevian triangle of a point with respect to the reference triangle.
Triangle$CevianTriangle(P)
Pa point
A Triangle object.
MalfattiCircles()
Malfatti circles of the triangle.
Triangle$MalfattiCircles(tangencyPoints = FALSE)
tangencyPointslogical, whether to retourn the tangency points of the Malfatti circles as an attribute.
A list with the three Malfatti circles, Circle objects.
t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2))
Mcircles <- t$MalfattiCircles(TRUE)
plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(0,2.5),
xlab = NA, ylab = NA)
grid()
draw(t, col = "blue", lwd = 2)
invisible(lapply(Mcircles, draw, col = "green", border = "red"))
invisible(lapply(attr(Mcircles, "tangencyPoints"), function(P){
points(P[1], P[2], pch = 19)
}))
AjimaMalfatti1()
First Ajima-Malfatti point of the triangle.
Triangle$AjimaMalfatti1()
AjimaMalfatti2()
Second Ajima-Malfatti point of the triangle.
Triangle$AjimaMalfatti2()
equalDetourPoint()
Equal detour point of the triangle.
Triangle$equalDetourPoint(detour = FALSE)
detourlogical, whether to return the detour as an attribute
Also known as the X(176) triangle center.
trilinearToPoint()
Point given by trilinear coordinates.
Triangle$trilinearToPoint(x, y, z)
x, y, ztrilinear coordinates
The point with trilinear coordinates x:y:z with respect to
the reference triangle.
t <- Triangle$new(c(0,0), c(2,1), c(5,7)) incircle <- t$incircle() t$trilinearToPoint(1, 1, 1) incircle$center
pointToTrilinear()
Give the trilinear coordinates of a point with respect to the reference triangle.
Triangle$pointToTrilinear(P)
Pa point
The trilinear coordinates, a numeric vector of length 3.
isogonalConjugate()
Isogonal conjugate of a point with respect to the reference triangle.
Triangle$isogonalConjugate(P)
Pa point
A point, the isogonal conjugate of P.
rotate()
Rotate the triangle.
Triangle$rotate(alpha, O, degrees = TRUE)
alphaangle of rotation
Ocenter of rotation
degreeslogical, whether alpha is given in degrees
A Triangle object.
translate()
Translate the triangle.
Triangle$translate(v)
vthe vector of translation
A Triangle object.
SteinerEllipse()
The Steiner ellipse (or circumellipse) of the reference triangle. This is the ellipse passing through the three vertices of the triangle and centered at the centroid of the triangle.
Triangle$SteinerEllipse()
An Ellipse object.
t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2))
ell <- t$SteinerEllipse()
plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.7,2.4),
xlab = NA, ylab = NA)
draw(t, col = "blue", lwd = 2)
draw(ell, border = "red", lwd =2)
SteinerInellipse()
The Steiner inellipse (or midpoint ellipse) of the reference triangle. This is the ellipse tangent to the sides of the triangle at their midpoints, and centered at the centroid of the triangle.
Triangle$SteinerInellipse()
An Ellipse object.
t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2))
ell <- t$SteinerInellipse()
plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.1,2.4),
xlab = NA, ylab = NA)
draw(t, col = "blue", lwd = 2)
draw(ell, border = "red", lwd =2)
MandartInellipse()
The Mandart inellipse of the reference triangle. This is the unique ellipse tangent to the triangle's sides at the contact points of its excircles
Triangle$MandartInellipse()
An Ellipse object.
randomPoints()
Random points on or in the reference triangle.
Triangle$randomPoints(n, where = "in")
nan integer, the desired number of points
where"in" to generate inside the triangle,
"on" to generate on the sides of the triangle
The generated points in a two columns matrix with n rows.
hexylTriangle()
Hexyl triangle.
Triangle$hexylTriangle()
plot()
Plot a Triangle object.
Triangle$plot(add = FALSE, ...)
addBoolean, whether to add the plot to the current plot
...named arguments passed to polygon
Nothing, called for plotting only.
trgl <- Triangle$new(c(0, 0), c(1, 0), c(0.5, sqrt(3)/2)) trgl$plot(col = "yellow", border = "red")
clone()
The objects of this class are cloneable with this method.
Triangle$clone(deep = FALSE)
deepWhether to make a deep clone.
The Steiner ellipse is also the smallest area ellipse which passes
through the vertices of the triangle, and thus can be obtained with
the function EllipseFromThreeBoundaryPoints. We can also
note that the major axis of the Steiner ellipse is the Deming
least squares line of the three triangle vertices.
TriangleThreeLines to define a triangle by three lines.
# incircle and excircles A <- c(0,0); B <- c(1,2); C <- c(3.5,1) t <- Triangle$new(A, B, C) incircle <- t$incircle() excircles <- t$excircles() JA <- excircles$A$center JB <- excircles$B$center JC <- excircles$C$center JAJBJC <- Triangle$new(JA, JB, JC) A_JA <- Line$new(A, JA, FALSE, FALSE) B_JB <- Line$new(B, JB, FALSE, FALSE) C_JC <- Line$new(C, JC, FALSE, FALSE) opar <- par(mar = c(0,0,0,0)) plot(NULL, asp = 1, xlim = c(0,6), ylim = c(-4,4), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(incircle, border = "orange") draw(excircles$A); draw(excircles$B); draw(excircles$C) draw(JAJBJC, col = "blue") draw(A_JA, col = "green") draw(B_JB, col = "green") draw(C_JC, col = "green") par(opar) ## ------------------------------------------------ ## Method `Triangle$new` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(1,0), c(1,1)) t t$C t$C <- c(2,2) t ## ------------------------------------------------ ## Method `Triangle$print` ## ------------------------------------------------ Triangle$new(c(0,0), c(1,0), c(1,1)) ## ------------------------------------------------ ## Method `Triangle$NagelTriangle` ## ------------------------------------------------ t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6)) lineAB <- Line$new(t$A, t$B) lineAC <- Line$new(t$A, t$C) lineBC <- Line$new(t$B, t$C) NagelTriangle <- t$NagelTriangle(NagelPoint = TRUE) NagelPoint <- attr(NagelTriangle, "Nagel point") excircles <- t$excircles() opar <- par(mar = c(0,0,0,0)) plot(0, 0, type="n", asp = 1, xlim = c(-1,5), ylim = c(-3,3), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(lineAB); draw(lineAC); draw(lineBC) draw(excircles$A, border = "orange") draw(excircles$B, border = "orange") draw(excircles$C, border = "orange") draw(NagelTriangle, lwd = 2, col = "red") draw(Line$new(t$A, NagelTriangle$A, FALSE, FALSE), col = "blue") draw(Line$new(t$B, NagelTriangle$B, FALSE, FALSE), col = "blue") draw(Line$new(t$C, NagelTriangle$C, FALSE, FALSE), col = "blue") points(rbind(NagelPoint), pch = 19) par(opar) ## ------------------------------------------------ ## Method `Triangle$symmedialTriangle` ## ------------------------------------------------ t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6)) symt <- t$symmedialTriangle() symmedianA <- Line$new(t$A, symt$A, FALSE, FALSE) symmedianB <- Line$new(t$B, symt$B, FALSE, FALSE) symmedianC <- Line$new(t$C, symt$C, FALSE, FALSE) K <- t$symmedianPoint() opar <- par(mar = c(0,0,0,0)) plot(NULL, asp = 1, xlim = c(-1,5), ylim = c(-3,3), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(symmedianA, lwd = 2, col = "blue") draw(symmedianB, lwd = 2, col = "blue") draw(symmedianC, lwd = 2, col = "blue") points(rbind(K), pch = 19, col = "red") par(opar) ## ------------------------------------------------ ## Method `Triangle$MalfattiCircles` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) Mcircles <- t$MalfattiCircles(TRUE) plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(0,2.5), xlab = NA, ylab = NA) grid() draw(t, col = "blue", lwd = 2) invisible(lapply(Mcircles, draw, col = "green", border = "red")) invisible(lapply(attr(Mcircles, "tangencyPoints"), function(P){ points(P[1], P[2], pch = 19) })) ## ------------------------------------------------ ## Method `Triangle$trilinearToPoint` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,1), c(5,7)) incircle <- t$incircle() t$trilinearToPoint(1, 1, 1) incircle$center ## ------------------------------------------------ ## Method `Triangle$SteinerEllipse` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) ell <- t$SteinerEllipse() plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.7,2.4), xlab = NA, ylab = NA) draw(t, col = "blue", lwd = 2) draw(ell, border = "red", lwd =2) ## ------------------------------------------------ ## Method `Triangle$SteinerInellipse` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) ell <- t$SteinerInellipse() plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.1,2.4), xlab = NA, ylab = NA) draw(t, col = "blue", lwd = 2) draw(ell, border = "red", lwd =2) ## ------------------------------------------------ ## Method `Triangle$plot` ## ------------------------------------------------ trgl <- Triangle$new(c(0, 0), c(1, 0), c(0.5, sqrt(3)/2)) trgl$plot(col = "yellow", border = "red")# incircle and excircles A <- c(0,0); B <- c(1,2); C <- c(3.5,1) t <- Triangle$new(A, B, C) incircle <- t$incircle() excircles <- t$excircles() JA <- excircles$A$center JB <- excircles$B$center JC <- excircles$C$center JAJBJC <- Triangle$new(JA, JB, JC) A_JA <- Line$new(A, JA, FALSE, FALSE) B_JB <- Line$new(B, JB, FALSE, FALSE) C_JC <- Line$new(C, JC, FALSE, FALSE) opar <- par(mar = c(0,0,0,0)) plot(NULL, asp = 1, xlim = c(0,6), ylim = c(-4,4), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(incircle, border = "orange") draw(excircles$A); draw(excircles$B); draw(excircles$C) draw(JAJBJC, col = "blue") draw(A_JA, col = "green") draw(B_JB, col = "green") draw(C_JC, col = "green") par(opar) ## ------------------------------------------------ ## Method `Triangle$new` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(1,0), c(1,1)) t t$C t$C <- c(2,2) t ## ------------------------------------------------ ## Method `Triangle$print` ## ------------------------------------------------ Triangle$new(c(0,0), c(1,0), c(1,1)) ## ------------------------------------------------ ## Method `Triangle$NagelTriangle` ## ------------------------------------------------ t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6)) lineAB <- Line$new(t$A, t$B) lineAC <- Line$new(t$A, t$C) lineBC <- Line$new(t$B, t$C) NagelTriangle <- t$NagelTriangle(NagelPoint = TRUE) NagelPoint <- attr(NagelTriangle, "Nagel point") excircles <- t$excircles() opar <- par(mar = c(0,0,0,0)) plot(0, 0, type="n", asp = 1, xlim = c(-1,5), ylim = c(-3,3), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(lineAB); draw(lineAC); draw(lineBC) draw(excircles$A, border = "orange") draw(excircles$B, border = "orange") draw(excircles$C, border = "orange") draw(NagelTriangle, lwd = 2, col = "red") draw(Line$new(t$A, NagelTriangle$A, FALSE, FALSE), col = "blue") draw(Line$new(t$B, NagelTriangle$B, FALSE, FALSE), col = "blue") draw(Line$new(t$C, NagelTriangle$C, FALSE, FALSE), col = "blue") points(rbind(NagelPoint), pch = 19) par(opar) ## ------------------------------------------------ ## Method `Triangle$symmedialTriangle` ## ------------------------------------------------ t <- Triangle$new(c(0,-2), c(0.5,1), c(3,0.6)) symt <- t$symmedialTriangle() symmedianA <- Line$new(t$A, symt$A, FALSE, FALSE) symmedianB <- Line$new(t$B, symt$B, FALSE, FALSE) symmedianC <- Line$new(t$C, symt$C, FALSE, FALSE) K <- t$symmedianPoint() opar <- par(mar = c(0,0,0,0)) plot(NULL, asp = 1, xlim = c(-1,5), ylim = c(-3,3), xlab = NA, ylab = NA, axes = FALSE) draw(t, lwd = 2) draw(symmedianA, lwd = 2, col = "blue") draw(symmedianB, lwd = 2, col = "blue") draw(symmedianC, lwd = 2, col = "blue") points(rbind(K), pch = 19, col = "red") par(opar) ## ------------------------------------------------ ## Method `Triangle$MalfattiCircles` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) Mcircles <- t$MalfattiCircles(TRUE) plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(0,2.5), xlab = NA, ylab = NA) grid() draw(t, col = "blue", lwd = 2) invisible(lapply(Mcircles, draw, col = "green", border = "red")) invisible(lapply(attr(Mcircles, "tangencyPoints"), function(P){ points(P[1], P[2], pch = 19) })) ## ------------------------------------------------ ## Method `Triangle$trilinearToPoint` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,1), c(5,7)) incircle <- t$incircle() t$trilinearToPoint(1, 1, 1) incircle$center ## ------------------------------------------------ ## Method `Triangle$SteinerEllipse` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) ell <- t$SteinerEllipse() plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.7,2.4), xlab = NA, ylab = NA) draw(t, col = "blue", lwd = 2) draw(ell, border = "red", lwd =2) ## ------------------------------------------------ ## Method `Triangle$SteinerInellipse` ## ------------------------------------------------ t <- Triangle$new(c(0,0), c(2,0.5), c(1.5,2)) ell <- t$SteinerInellipse() plot(NULL, asp = 1, xlim = c(0,2.5), ylim = c(-0.1,2.4), xlab = NA, ylab = NA) draw(t, col = "blue", lwd = 2) draw(ell, border = "red", lwd =2) ## ------------------------------------------------ ## Method `Triangle$plot` ## ------------------------------------------------ trgl <- Triangle$new(c(0, 0), c(1, 0), c(0.5, sqrt(3)/2)) trgl$plot(col = "yellow", border = "red")
Return the triangle formed by three lines.
TriangleThreeLines(line1, line2, line3)TriangleThreeLines(line1, line2, line3)
line1, line2, line3
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A Triangle object.
Circle centered at the origin with radius 1.
unitCircleunitCircle
An object of class Circle (inherits from R6) of length 25.