Inversion: R6 class representing an inversion
In PlaneGeometry: Plane Geometry
Inversion R Documentation
R6 class representing an inversion
Description
An inversion is given by a pole (a point) and a power (a number,
possibly negative, but not zero).
Active bindings
poleget or set the pole
powerget or set the power
Methods
Public methods
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Method new()
Create a new Inversion object.
Usage
Inversion$new(pole, power)
Arguments
polethe pole
powerthe power
Returns
A new Inversion object.
Method print()
Show instance of an inversion object.
Usage
Inversion$print(...)
Arguments
...ignored
Examples
Inversion$new(c(0,0), 2)
Method invert()
Inversion of a point.
Usage
Inversion$invert(M)
Arguments
Ma point or Inf
Returns
A point or Inf, the image of M.
Method transform()
An alias of invert.
Usage
Inversion$transform(M)
Arguments
Ma point or Inf
Returns
A point or Inf, the image of M.
Method invertCircle()
Inversion of a circle.
Usage
Inversion$invertCircle(circ)
Arguments
circa Circle object
Returns
A Circle object or a Line object.
Examples
# 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 transformCircle()
An alias of invertCircle.
Usage
Inversion$transformCircle(circ)
Arguments
circa Circle object
Returns
A Circle object or a Line object.
Method invertLine()
Inversion of a line.
Usage
Inversion$invertLine(line)
Arguments
linea Line object
Returns
A Circle object or a Line object.
Method transformLine()
An alias of invertLine.
Usage
Inversion$transformLine(line)
Arguments
linea Line object
Returns
A Circle object or a Line object.
Method invertGcircle()
Inversion of a generalized circle (i.e. a circle or a line).
Usage
Inversion$invertGcircle(gcircle)
Arguments
gcirclea Circle object or a Line object
Returns
A Circle object or a Line object.
Method compose()
Compose the reference inversion with another inversion.
The result is a Möbius transformation.
Usage
Inversion$compose(iota1, left = TRUE)
Arguments
iota1an Inversion object
leftlogical, whether to compose at left or at right (i.e.
returns iota1 o iota0 or iota0 o iota1)
Returns
A Mobius object.
Method clone()
The objects of this class are cloneable with this method.
Usage
Inversion$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
inversionSwappingTwoCircles,
inversionFixingTwoCircles,
inversionFixingThreeCircles to create some inversions.
Examples
## ------------------------------------------------
## 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$invertCircle(circ)
draw(iotacirc)
}
par(opar)
PlaneGeometry documentation built on Aug. 10, 2023, 1:09 a.m.
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| Inversion | R Documentation |
R6 class representing an inversion
Description
An inversion is given by a pole (a point) and a power (a number, possibly negative, but not zero).
Active bindings
poleget or set the pole
powerget or set the power
Methods
Public methods
Method new()
Create a new Inversion object.
Usage
Inversion$new(pole, power)
Arguments
polethe pole
powerthe power
Returns
A new Inversion object.
Method print()
Show instance of an inversion object.
Usage
Inversion$print(...)
Arguments
...ignored
Examples
Inversion$new(c(0,0), 2)
Method invert()
Inversion of a point.
Usage
Inversion$invert(M)
Arguments
Ma point or
Inf
Returns
A point or Inf, the image of M.
Method transform()
An alias of invert.
Usage
Inversion$transform(M)
Arguments
Ma point or
Inf
Returns
A point or Inf, the image of M.
Method invertCircle()
Inversion of a circle.
Usage
Inversion$invertCircle(circ)
Arguments
circa
Circleobject
Returns
A Circle object or a Line object.
Examples
# A Pappus chain
# https://www.cut-the-knot.org/Curriculum/Geometry/InversionInArbelos.shtml
opar <- par(mar = c(0,0,0,0))
plot(0, 0, type = "n", asp = 1, xlim = c(0,6), ylim = c(-4,4),
xlab = NA, ylab = NA, axes = FALSE)
A <- c(0,0); B <- c(6,0)
ABsqr <- c(crossprod(A-B))
iota <- Inversion$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$invertCircle(circ)
draw(iotacirc)
}
par(opar)
Method transformCircle()
An alias of invertCircle.
Usage
Inversion$transformCircle(circ)
Arguments
circa
Circleobject
Returns
A Circle object or a Line object.
Method invertLine()
Inversion of a line.
Usage
Inversion$invertLine(line)
Arguments
linea
Lineobject
Returns
A Circle object or a Line object.
Method transformLine()
An alias of invertLine.
Usage
Inversion$transformLine(line)
Arguments
linea
Lineobject
Returns
A Circle object or a Line object.
Method invertGcircle()
Inversion of a generalized circle (i.e. a circle or a line).
Usage
Inversion$invertGcircle(gcircle)
Arguments
gcirclea
Circleobject or aLineobject
Returns
A Circle object or a Line object.
Method compose()
Compose the reference inversion with another inversion. The result is a Möbius transformation.
Usage
Inversion$compose(iota1, left = TRUE)
Arguments
iota1an
Inversionobjectleftlogical, whether to compose at left or at right (i.e. returns
iota1 o iota0oriota0 o iota1)
Returns
A Mobius object.
Method clone()
The objects of this class are cloneable with this method.
Usage
Inversion$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
inversionSwappingTwoCircles,
inversionFixingTwoCircles,
inversionFixingThreeCircles to create some inversions.
Examples
## ------------------------------------------------
## Method `Inversion$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)
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