straight_line_trajectory

class flatsurf.geometry.straight_line_trajectory.AbstractStraightLineTrajectory

coding(alphabet=None)

Return the coding of this trajectory with respect to the sides of the polygons

INPUT:

• alphabet – an optional dictionary (lab,nb) -> letter. If some labels are avoided then these crossings are ignored.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: t = translation_surfaces.square_torus()

sage: v = t.tangent_vector(0, (1/2,0), (5,6))
sage: l = v.straight_line_trajectory()
sage: alphabet = {(0,0): 'a', (0,1): 'b', (0,2):'a', (0,3): 'b'}
sage: l.coding()
[(0, 0), (0, 1)]
sage: l.coding(alphabet)
['a', 'b']
sage: l.flow(10); l.flow(-10)
sage: l.coding()
[(0, 2), (0, 1), (0, 2), (0, 1), (0, 2), (0, 1), (0, 2), (0, 1), (0, 2)]
sage: print(''.join(l.coding(alphabet)))
ababababa

sage: v = t.tangent_vector(0, (1/2,0), (7,13))
sage: l = v.straight_line_trajectory()
sage: l.flow(10); l.flow(-10)
sage: print(''.join(l.coding(alphabet)))
aabaabaababaabaabaab

For a closed trajectory, the last label (corresponding also to the starting point) is not considered:

sage: v = t.tangent_vector(0, (1/5,1/7), (1,1))
sage: l = v.straight_line_trajectory()
sage: l.flow(10)
sage: l.is_closed()
True
sage: l.coding(alphabet)
['a', 'b']

Check that the saddle connections that are obtained in the torus get the expected coding:

sage: for _ in range(10):  # long time (.6s)
....:     x = ZZ.random_element(1,30)
....:     y = ZZ.random_element(1,30)
....:     x,y = x/gcd(x,y), y/gcd(x,y)
....:     v = t.tangent_vector(0, (0,0), (x,y))
....:     l = v.straight_line_trajectory()
....:     l.flow(200); l.flow(-200)
....:     w = ''.join(l.coding(alphabet))
....:     assert Word(w+'ab'+w).is_balanced()
....:     assert Word(w+'ba'+w).is_balanced()
....:     assert w.count('a') == y-1
....:     assert w.count('b') == x-1

combinatorial_length()

cylinder()

If this is a closed orbit, return the associated maximal cylinder. Raises a ValueError if this trajectory is not closed.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.regular_octagon()
sage: v = s.tangent_vector(0,(1/2,0),(sqrt(2),1))
sage: traj = v.straight_line_trajectory()
sage: traj.flow(4)
sage: traj.is_closed()
True
sage: cyl = traj.cylinder()
sage: cyl.area()                # a = sqrt(2)
a + 1
sage: cyl.holonomy()
(3*a + 4, 2*a + 3)
sage: cyl.edges()
(2, 3, 3, 2, 4)

graphical_trajectory(graphical_surface=None, **options)

Returns a GraphicalStraightLineTrajectory corresponding to this trajectory in the provided GraphicalSurface.

initial_tangent_vector()

intersections(traj, count_singularities=False, include_segments=False)

Return the set of SurfacePoints representing the intersections of this trajectory with the provided trajectory or SaddleConnection.

Singularities will be included only if count_singularities is set to True.

If include_segments is True, it iterates over triples consisting of the SurfacePoint, and two sets. The first set consists of segments of this trajectory that contain the point and the second set consists of segments of traj that contain the point.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.square_torus()
sage: traj1 = s.tangent_vector(0,(1/2,0),(1,1)).straight_line_trajectory()
sage: traj1.flow(3)
sage: traj1.is_closed()
True
sage: traj2 = s.tangent_vector(0,(1/2,0),(-1,1)).straight_line_trajectory()
sage: traj2.flow(3)
sage: traj2.is_closed()
True
sage: sum(1 for _ in traj1.intersections(traj2))
2

sage: for p, (segs1, segs2) in traj1.intersections(traj2, include_segments=True):
....:     print(p)
....:     print(len(segs1), len(segs2))
Point (1/2, 0) of polygon 0
2 2
Point (0, 1/2) of polygon 0
2 2

intersects(traj, count_singularities=False)

Return true if this trajectory intersects the other trajectory.

is_closed()

plot(*args, **options)

Plot this trajectory by converting to a graphical trajectory.

If any arguments are provided in \(*args\) it must be only one argument containing a GraphicalSurface. The keyword arguments in \(**options\) are passed on to flatsurf.graphical.straight_line_trajectory.GraphicalStraightLineTrajectory.plot().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.square_torus()
sage: v = T.tangent_vector(0, (0,0), (5,7))
sage: L = v.straight_line_trajectory()
sage: L.plot()
...Graphics object consisting of 1 graphics primitive

sage: L.plot(color='red')
...Graphics object consisting of 1 graphics primitive

segment(i)

segments()

surface()

terminal_tangent_vector()

class flatsurf.geometry.straight_line_trajectory.SegmentInPolygon(start, end=None)

Maximal segment in a polygon of a similarity surface

EXAMPLES:

sage: from flatsurf import similarity_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import SegmentInPolygon
sage: s = similarity_surfaces.example()
sage: v = s.tangent_vector(0, (1/3,-1/4), (0,1))
sage: SegmentInPolygon(v)
Segment in polygon 0 starting at (1/3, -1/3) and ending at (1/3, 0)

edge()

end()

Return a TangentVector associated to the end of a trajectory, pointed backward.

end_is_singular()

invert()

is_edge()

next()

Return the next segment obtained by continuing straight through the end point.

EXAMPLES:

sage: from flatsurf import similarity_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import SegmentInPolygon

sage: s = similarity_surfaces.example()
sage: s.polygon(0)
Polygon(vertices=[(0, 0), (2, -2), (2, 0)])
sage: s.polygon(1)
Polygon(vertices=[(0, 0), (2, 0), (1, 3)])
sage: v = s.tangent_vector(0, (0,0), (3,-1))
sage: seg = SegmentInPolygon(v)
sage: seg
Segment in polygon 0 starting at (0, 0) and ending at (2, -2/3)
sage: seg.next()
Segment in polygon 1 starting at (2/3, 2) and ending at (14/9, 4/3)

polygon_label()

previous()

start()

Return the tangent vector associated to the start of a trajectory pointed forward.

start_is_singular()

class flatsurf.geometry.straight_line_trajectory.StraightLineTrajectory(tangent_vector)

Straight-line trajectory in a similarity surface.

EXAMPLES:

# Demonstrate the handling of edges
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import StraightLineTrajectory
sage: p = SymmetricGroup(2)('(1,2)')
sage: s = translation_surfaces.origami(p,p)
sage: traj = StraightLineTrajectory(s.tangent_vector(1,(0,0),(1,0)))
sage: traj
Straight line trajectory made of 1 segments from (0, 0) in polygon 1 to (1, 1) in polygon 2
sage: traj.is_saddle_connection()
True
sage: traj2 = StraightLineTrajectory(s.tangent_vector(1,(0,0),(0,1)))
sage: traj2
Straight line trajectory made of 1 segments from (1, 0) in polygon 2 to (0, 1) in polygon 1
sage: traj2.is_saddle_connection()
True

combinatorial_length()

flow(steps)

Append or prepend segments to the trajectory. If steps is positive, attempt to append this many segments. If steps is negative, attempt to prepend this many segments. Will fail gracefully the trajectory hits a singularity or closes up.

EXAMPLES:

sage: from flatsurf import similarity_surfaces

sage: s = similarity_surfaces.example()
sage: v = s.tangent_vector(0, (1,-1/2), (3,-1))
sage: traj = v.straight_line_trajectory()
sage: traj
Straight line trajectory made of 1 segments from (1/4, -1/4) in polygon 0 to (2, -5/6) in polygon 0
sage: traj.flow(1)
sage: traj
Straight line trajectory made of 2 segments from (1/4, -1/4) in polygon 0 to (61/36, 11/12) in polygon 1
sage: traj.flow(-1)
sage: traj
Straight line trajectory made of 3 segments from (15/16, 45/16) in polygon 1 to (61/36, 11/12) in polygon 1

is_backward_separatrix()

is_closed()

Test whether this is a closed trajectory.

By convention, by a closed trajectory we mean a trajectory without any singularities.

See also

is_saddle_connection()

EXAMPLES:

An example in a cone surface covered by the torus:

sage: from flatsurf import MutableOrientedSimilaritySurface, polygons
sage: p = polygons.square()
sage: s = MutableOrientedSimilaritySurface(p.base_ring())
sage: s.add_polygon(p)
0
sage: s.glue((0, 0), (0, 3))
sage: s.glue((0, 1), (0, 2))
sage: s.set_immutable()
sage: t = s

sage: v = t.tangent_vector(0, (1/2,0), (1/3,7/5))
sage: l = v.straight_line_trajectory()
sage: l.is_closed()
False
sage: l.flow(100)
sage: l.is_closed()
True

sage: v = t.tangent_vector(0, (1/2,0), (1/3,2/5))
sage: l = v.straight_line_trajectory()
sage: l.flow(100)
sage: l.is_closed()
False
sage: l.is_saddle_connection()
False
sage: l.flow(-100)
sage: l.is_saddle_connection()
True

is_forward_separatrix()

is_saddle_connection()

segment(i)

EXAMPLES:

sage: from flatsurf import translation_surfaces

sage: O = translation_surfaces.regular_octagon()
sage: v = O.tangent_vector(0, (1,1), (33,45))
sage: L = v.straight_line_trajectory()
sage: L.segment(0)
Segment in polygon 0 starting at (4/15, 0) and ending at (11/26*a +
1, 15/26*a + 1)
sage: L.flow(-1)
sage: L.segment(0)
Segment in polygon 0 starting at (-1/2*a, 7/22*a + 7/11) and ending
at (4/15, a + 1)
sage: L.flow(1)
sage: L.segment(2)
Segment in polygon 0 starting at (-1/13*a, 1/13*a) and ending at
(9/26*a + 11/13, 17/26*a + 15/13)

segments()

surface()

class flatsurf.geometry.straight_line_trajectory.StraightLineTrajectoryTranslation(tangent_vector)

Straight line trajectory in a translation surface.

This is similar to StraightLineTrajectory but implemented using interval exchange maps. It should be faster than the implementation via segments and flowing in polygons.

This class only stores a list of triples (p, e, x) where:

• p is a label of a polygon

• e is the number of some edge in p

• x is the position of the point in e (be careful that it is not necessarily a number between 0 and 1. It is given relatively to the length of the induced interval in the iet)

combinatorial_length()

flow(steps)

is_backward_separatrix()

is_closed()

is_forward_separatrix()

is_saddle_connection()

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import StraightLineTrajectoryTranslation

sage: torus = translation_surfaces.square_torus()
sage: v = torus.tangent_vector(0, (1/2,1/2), (1,1))
sage: S = StraightLineTrajectoryTranslation(v)
sage: S.is_saddle_connection()
True

sage: v = torus.tangent_vector(0, (1/3,2/3), (1,2))
sage: S = StraightLineTrajectoryTranslation(v)
sage: S.is_saddle_connection()
False
sage: S.flow(1)
sage: S.is_saddle_connection()
True

segment(i)

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import StraightLineTrajectoryTranslation

sage: O = translation_surfaces.regular_octagon()
sage: v = O.tangent_vector(0, (1,1), (33,45))
sage: L = StraightLineTrajectoryTranslation(v)
sage: L.segment(0)
Segment in polygon 0 starting at (4/15, 0) and ending at (11/26*a +
1, 15/26*a + 1)
sage: L.flow(-1)
sage: L.segment(0)
Segment in polygon 0 starting at (-1/2*a, 7/22*a + 7/11) and ending
at (4/15, a + 1)
sage: L.flow(1)
sage: L.segment(2)
Segment in polygon 0 starting at (-1/13*a, 1/13*a) and ending at
(9/26*a + 11/13, 17/26*a + 15/13)

segments()

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.straight_line_trajectory import StraightLineTrajectoryTranslation

sage: s = translation_surfaces.square_torus()
sage: v = s.tangent_vector(0, (0,0), (1,1+AA(5).sqrt()), ring=AA)
sage: L = StraightLineTrajectoryTranslation(v)
sage: L.flow(2)
sage: L.segments()
[Segment in polygon 0 starting at (0, 0) and ending at (0.3090169943749474?, 1),
 Segment in polygon 0 starting at (0.3090169943749474?, 0) and ending at (0.618033988749895?, 1),
 Segment in polygon 0 starting at (0.618033988749895?, 0) and ending at (0.9270509831248423?, 1)]

flatsurf.geometry.straight_line_trajectory.get_linearity_coeff(u, v)

Given the two 2-dimensional vectors u and v, return a so that v = a*u

If the vectors are not colinear, a ValueError is raised.

EXAMPLES:

sage: from flatsurf.geometry.straight_line_trajectory import get_linearity_coeff

sage: V = VectorSpace(QQ,2)
sage: get_linearity_coeff(V((1,0)), V((2,0)))
2
sage: get_linearity_coeff(V((2,0)), V((1,0)))
1/2
sage: get_linearity_coeff(V((0,1)), V((0,2)))
2
sage: get_linearity_coeff(V((0,2)), V((0,1)))
1/2
sage: get_linearity_coeff(V((1,2)), V((-2,-4)))
-2

sage: get_linearity_coeff(V((1,1)), V((-1,1)))
Traceback (most recent call last):
...
ValueError: non colinear