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PlaneGeometry (version 1.6.0)

Inversion: 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).

Arguments

Active bindings

pole

get or set the pole

power

get or set the power

Methods

Method new()

Create a new Inversion object.

Usage

Inversion$new(pole, power)

Arguments

pole

the pole

power

the 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

M

a point or Inf

Returns

A point or Inf, the image of M.

Method transform()

An alias of invert.

Usage

Inversion$transform(M)

Arguments

M

a point or Inf

Returns

A point or Inf, the image of M.

Method invertCircle()

Inversion of a circle.

Usage

Inversion$invertCircle(circ)

Arguments

circ

a 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

circ

a Circle object

Returns

A Circle object or a Line object.

Method invertLine()

Inversion of a line.

Usage

Inversion$invertLine(line)

Arguments

line

a Line object

Returns

A Circle object or a Line object.

Method transformLine()

An alias of invertLine.

Usage

Inversion$transformLine(line)

Arguments

line

a 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

gcircle

a 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

iota1

an Inversion object

left

logical, 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

deep

Whether to make a deep clone.

See Also

inversionSwappingTwoCircles, inversionFixingTwoCircles, inversionFixingThreeCircles to create some inversions.

Examples

Run this code

## ------------------------------------------------
## 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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