MoebiusTransform F¶

moebius.spad line 1 [edit on github]

F: Field

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: from CoercibleTo 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.

CoercibleTo OutputForm