# MoebiusTransform FΒΆ

MoebiusTransform(`F`) is the domain of fractional linear (Moebius) transformations over `F`.

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

/: (%, %) -> %

from Group

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> %

from Group

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm
commutator: (%, %) -> %

from Group

conjugate: (%, %) -> %

from Group

eval: (%, F) -> F

`eval(m, x)` returns `(a*x + b)/(c*x + d)` where `m = moebius(a, b, c, d)` (see moebius).

eval: (%, OnePointCompletion F) -> OnePointCompletion F

`eval(m, x)` returns `(a*x + b)/(c*x + d)` where `m = moebius(a, b, c, d)` (see moebius).

inv: % -> %

from Group

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

moebius: (F, F, F, F) -> %

`moebius(a, b, c, d)` returns `matrix [[a, b], [c, d]]`.

one?: % -> Boolean

from MagmaWithUnit

recip: % -> %

`recip(m)` = recip() * `m`

recip: % -> Union(%, failed)

from MagmaWithUnit

recip: () -> %

`recip()` returns `matrix [[0, 1], [1, 0]]` representing the map `x -> 1 / x`.

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from MagmaWithUnit

scale: (%, F) -> %

`scale(m, h)` returns `scale(h) * m` (see shift).

scale: F -> %

`scale(k)` returns `matrix [[k, 0], [0, 1]]` representing the map `x -> k * x`.

shift: (%, F) -> %

`shift(m, h)` returns `shift(h) * m` (see shift).

shift: F -> %

`shift(k)` returns `matrix [[1, k], [0, 1]]` representing the map `x -> x + k`.

BasicType

Group

Magma

MagmaWithUnit

Monoid

SemiGroup

SetCategory

TwoSidedRecip

unitsKnown