Mobius: R6 class representing a Möbius transformation.
In PlaneGeometry: Plane Geometry
Mobius R Documentation
R6 class representing a Möbius transformation.
Description
A Möbius transformation is given by a matrix of complex numbers
with non-null determinant.
Active bindings
aget or set a
bget or set b
cget or set c
dget or set d
Methods
Public methods
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Method new()
Create a new Mobius object.
Usage
Mobius$new(M)
Arguments
Mthe matrix corresponding to the Möbius transformation
Returns
A new Mobius object.
Method print()
Show instance of a Mobius object.
Usage
Mobius$print(...)
Arguments
...ignored
Examples
Mobius$new(rbind(c(1+1i,2),c(0,3-2i)))
Method getM()
Get the matrix corresponding to the Möbius transformation.
Usage
Mobius$getM()
Method compose()
Compose the reference Möbius transformation with another
Möbius transformation
Usage
Mobius$compose(M1, left = TRUE)
Arguments
M1a Mobius object
leftlogical, whether to compose at left or at right (i.e.
returns M1 o M0 or M0 o M1)
Returns
A Mobius object.
Method inverse()
Inverse of the reference Möbius transformation.
Usage
Mobius$inverse()
Returns
A Mobius object.
Method power()
Power of the reference Möbius transformation.
Usage
Mobius$power(k)
Arguments
kan integer, possibly negative
Returns
The Möbius transformation M^k,
where M is the reference Möbius transformation.
Method gpower()
Generalized power of the reference Möbius transformation.
Usage
Mobius$gpower(k)
Arguments
ka real number, possibly negative
Returns
A Mobius object, the generalized k-th power of
the reference Möbius transformation.
Examples
M <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i)))
Mroot <- M$gpower(1/2)
Mroot$compose(Mroot) # should be M
Method transform()
Transformation of a point by the reference Möbius transformation.
Usage
Mobius$transform(M)
Arguments
Ma point or Inf
Returns
A point or Inf, the image of M.
Examples
Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i)))
Mob$transform(c(1,1))
Mob$transform(Inf)
Method fixedPoints()
Returns the fixed points of the reference Möbius transformation.
Usage
Mobius$fixedPoints()
Returns
One point, or a list of two points, or a message in the case when the transformation is the identity map.
Method transformCircle()
Transformation of a circle by the reference Möbius transformation.
Usage
Mobius$transformCircle(circ)
Arguments
circa Circle object
Returns
A Circle object or a Line object.
Method transformLine()
Transformation of a line by the reference Möbius transformation.
Usage
Mobius$transformLine(line)
Arguments
linea Line object
Returns
A Circle object or a Line object.
Method transformGcircle()
Transformation of a generalized circle (i.e. a circle or a
line) by the reference Möbius transformation.
Usage
Mobius$transformGcircle(gcirc)
Arguments
gcirca Circle object or a Line object
Returns
A Circle object or a Line object.
Method clone()
The objects of this class are cloneable with this method.
Usage
Mobius$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
MobiusMappingThreePoints to create a Möbius
transformation, and also the compose method of the
Inversion R6 class.
Examples
## ------------------------------------------------
## 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)
PlaneGeometry documentation built on Aug. 10, 2023, 1:09 a.m.
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| Mobius | R Documentation |
R6 class representing a Möbius transformation.
Description
A Möbius transformation is given by a matrix of complex numbers with non-null determinant.
Active bindings
aget or set
abget or set
bcget or set
cdget or set
d
Methods
Public methods
Method new()
Create a new Mobius object.
Usage
Mobius$new(M)
Arguments
Mthe matrix corresponding to the Möbius transformation
Returns
A new Mobius object.
Method print()
Show instance of a Mobius object.
Usage
Mobius$print(...)
Arguments
...ignored
Examples
Mobius$new(rbind(c(1+1i,2),c(0,3-2i)))
Method getM()
Get the matrix corresponding to the Möbius transformation.
Usage
Mobius$getM()
Method compose()
Compose the reference Möbius transformation with another Möbius transformation
Usage
Mobius$compose(M1, left = TRUE)
Arguments
M1a
Mobiusobjectleftlogical, whether to compose at left or at right (i.e. returns
M1 o M0orM0 o M1)
Returns
A Mobius object.
Method inverse()
Inverse of the reference Möbius transformation.
Usage
Mobius$inverse()
Returns
A Mobius object.
Method power()
Power of the reference Möbius transformation.
Usage
Mobius$power(k)
Arguments
kan integer, possibly negative
Returns
The Möbius transformation M^k,
where M is the reference Möbius transformation.
Method gpower()
Generalized power of the reference Möbius transformation.
Usage
Mobius$gpower(k)
Arguments
ka real number, possibly negative
Returns
A Mobius object, the generalized k-th power of
the reference Möbius transformation.
Examples
M <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mroot <- M$gpower(1/2) Mroot$compose(Mroot) # should be M
Method transform()
Transformation of a point by the reference Möbius transformation.
Usage
Mobius$transform(M)
Arguments
Ma point or
Inf
Returns
A point or Inf, the image of M.
Examples
Mob <- Mobius$new(rbind(c(1+1i,2),c(0,3-2i))) Mob$transform(c(1,1)) Mob$transform(Inf)
Method fixedPoints()
Returns the fixed points of the reference Möbius transformation.
Usage
Mobius$fixedPoints()
Returns
One point, or a list of two points, or a message in the case when the transformation is the identity map.
Method transformCircle()
Transformation of a circle by the reference Möbius transformation.
Usage
Mobius$transformCircle(circ)
Arguments
circa
Circleobject
Returns
A Circle object or a Line object.
Method transformLine()
Transformation of a line by the reference Möbius transformation.
Usage
Mobius$transformLine(line)
Arguments
linea
Lineobject
Returns
A Circle object or a Line object.
Method transformGcircle()
Transformation of a generalized circle (i.e. a circle or a line) by the reference Möbius transformation.
Usage
Mobius$transformGcircle(gcirc)
Arguments
gcirca
Circleobject or aLineobject
Returns
A Circle object or a Line object.
Method clone()
The objects of this class are cloneable with this method.
Usage
Mobius$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
See Also
MobiusMappingThreePoints to create a Möbius
transformation, and also the compose method of the
Inversion R6 class.
Examples
## ------------------------------------------------
## 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)
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