`similarity`¶

Initialize self. See help(type(self)) for accurate signature.

class flatsurf.geometry.similarity.Similarity(p, a, b, s, t, sign)[source]¶

Class for a similarity of the plane with possible reflection.

Construct the similarity (x,y) mapsto (ax-by+s,bx+ay+t) if sign=1, and (ax+by+s,bx-ay+t) if sign=-1

Return the 2x2 matrix corresponding to the derivative of this element

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)

sage: S((1,-2/3,1,1,-1)).derivative()
[   1 -2/3]
[-2/3   -1]

det()[source]¶: Return the determinant of this element

is_half_translation()[source]¶

Return whether this element is a half translation.

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)
sage: S((1,2)).is_half_translation()
True
sage: S((-1, 0, 0, 2)).is_half_translation()
True
sage: S((0,1,0,0)).is_half_translation()
False

is_isometry()[source]¶

Check whether this element is an isometry

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)
sage: S.one().is_isometry()
True
sage: S((0,1,0,0)).is_isometry()
True
sage: S((0,1,0,0,-1)).is_isometry()
True
sage: S((1,1,0,0)).is_isometry()
False
sage: S((3,-1/2)).is_isometry()
True

is_orientable()[source]¶

is_rotation()[source]¶

Check whether this element is a rotation

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)
sage: S((1,2)).is_rotation()
False
sage: S((0,-1,0,0)).is_rotation()
True
sage: S.one().is_rotation()
True

is_translation()[source]¶

Return whether this element is a translation.

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)
sage: S((1,2)).is_translation()
True
sage: S((1,0,3,-1/2)).is_translation()
True
sage: S((0,1,0,0)).is_translation()
False

matrix()[source]¶

Return the 3x3 matrix representative of this element

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: S = SimilarityGroup(QQ)

sage: S((1,-2/3,1,1,-1)).matrix()
[   1 -2/3    1]
[-2/3   -1    1]
[   0    0    1]

sign()[source]¶

class flatsurf.geometry.similarity.SimilarityGroup(base_ring)[source]¶

The group of possibly orientation reversing similarities in the plane.

This is the group generated by rotations, translations and dilations.

Element¶: alias of Similarity

base_ring()[source]¶

is_abelian()[source]¶

one()[source]¶

EXAMPLES:

sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: SimilarityGroup(QQ).one()
(x, y) |-> (x, y)
sage: SimilarityGroup(QQ).one().is_one()
True

flatsurf.geometry.similarity.similarity_from_vectors(u, v, matrix_space=None)[source]¶

Return the unique similarity matrix that maps u to v.

EXAMPLES:

sage: from flatsurf.geometry.similarity import similarity_from_vectors

sage: V = VectorSpace(QQ,2)
sage: u = V((1,0))
sage: v = V((0,1))
sage: m = similarity_from_vectors(u,v); m
[ 0 -1]
[ 1  0]
sage: m*u == v
True

sage: u = V((2,1))
sage: v = V((1,-2))
sage: m = similarity_from_vectors(u,v); m
[ 0  1]
[-1  0]
sage: m * u == v
True

An example built from the Pythagorean triple 3^2 + 4^2 = 5^2:

sage: u2 = V((5,0))
sage: v2 = V((3,4))
sage: m = similarity_from_vectors(u2,v2); m
[ 3/5 -4/5]
[ 4/5  3/5]
sage: m * u2 == v2
True

Some test over number fields:

sage: K.<sqrt2> = NumberField(x^2-2, embedding=1.4142)
sage: V = VectorSpace(K,2)
sage: u = V((sqrt2,0))
sage: v = V((1, 1))
sage: m = similarity_from_vectors(u,v); m
[ 1/2*sqrt2 -1/2*sqrt2]
[ 1/2*sqrt2  1/2*sqrt2]
sage: m*u == v
True

sage: m = similarity_from_vectors(u, 2*v); m
[ sqrt2 -sqrt2]
[ sqrt2  sqrt2]
sage: m*u == 2*v
True

similarity¶

`similarity`¶