Source code for flatsurf.geometry.interval_exchange_transformation

from sage.structure.sage_object import SageObject


[docs] class FlowPolygonMap(SageObject): r""" The map obtained as the return map of the flow on the sides of a (convex) polygon. Formally, this can be defined as follows: one start with two partition into (finitely many) intervals of a given interval. The map corresponds to changing from the first partition to the second. In other words, points are identified as pairs ``(i, x)`` where ``i`` is an atom and ``x`` is the relative position in this atom. Contrarily to an interval exchange transformation, things here are going from bottom to top. Note that this could also be used for homothetic surfaces. And to some extent to half translation surface. EXAMPLES:: sage: from flatsurf.geometry.interval_exchange_transformation import FlowPolygonMap sage: T = FlowPolygonMap(QQ, [0,1,2], [2,3,1], [2,1,0], [1,3,2]) sage: [T.forward_image(0,x) for x in range(3)] [(2, 0), (1, 0), (1, 1)] sage: [T.forward_image(1,x) for x in range(4)] [(1, 1), (1, 2), (0, 0), (0, 1)] sage: [T.forward_image(2,x) for x in range(1)] [(0, 1)] """ def __init__(self, ring, bot_labels, bot_lengths, top_labels, top_lengths): r""" INPUT: - ``ring`` -- the base ring for the lengths of the interval - ``bot_labels`` -- labels for the bottom partition - ``bot_lengths`` -- lengths for the bottom partition - ``top_labels`` -- labels for the top partition - ``top_lengths`` -- lengths for top partition """ if not all(x > ring.zero() for x in bot_lengths): raise ValueError if not all(x > ring.zero() for x in top_lengths): raise ValueError if len(bot_labels) != len(bot_lengths): raise ValueError if len(top_labels) != len(top_lengths): raise ValueError if sum(top_lengths) != sum(bot_lengths): raise ValueError self._ring = ring self._bot_labels = bot_labels self._top_labels = top_labels self._bot_labels_to_index = {j: i for i, j in enumerate(bot_labels)} self._top_labels_to_index = {j: i for i, j in enumerate(top_labels)} if len(self._bot_labels) != len(self._bot_labels_to_index): raise ValueError("non unique labels for bot: {}".format(bot_labels)) if len(self._top_labels) != len(self._top_labels_to_index): raise ValueError("non unique labels in top: {}".format(top_labels)) self._bot_lengths = list(map(ring, bot_lengths)) self._top_lengths = list(map(ring, top_lengths)) # forward image of intervals it = 0 lt = x1 = top_lengths[it] self._forward_images = [] for ib in range(len(bot_lengths)): lenb = bot_lengths[ib] self._forward_images.append((it, lt - x1)) while lenb and lenb >= x1: lenb -= x1 it += 1 if it < len(top_lengths): lt = x1 = top_lengths[it] else: lt = ring.zero() if lenb: x1 -= lenb # backward image of intervals ib = 0 lb = x1 = bot_lengths[ib] self._backward_images = [] for it in range(len(top_lengths)): lent = top_lengths[it] self._backward_images.append((ib, lb - x1)) while lent and lent >= x1: lent -= x1 ib += 1 if ib < len(bot_lengths): lb = x1 = bot_lengths[ib] else: lb = ring.zero() if lent: x1 -= lent
[docs] def length_bot(self, i): i = self._bot_labels_to_index[i] return self._bot_lengths[i]
[docs] def length_top(self, i): i = self._top_labels_to_index[i] return self._top_lengths[i]
def _repr_(self): s = ["Flow polygon map:"] s.append(" " + " ".join(str(x) for x in self._top_labels)) s.append(" " + " ".join(str(x) for x in self._bot_labels)) s.append("top lengths: {}".format(self._top_lengths)) s.append("bot lengths: {}".format(self._bot_lengths)) return "\n".join(s)
[docs] def forward_image(self, i, x): r""" Return the forward image. EXAMPLES:: sage: from flatsurf.geometry.interval_exchange_transformation import FlowPolygonMap Singularities are always sent to a ``(i,0)``:: sage: T = FlowPolygonMap(QQ, [0,1], [2,1], [2,3,4], [1,1,1]) sage: T.forward_image(0, 0) (2, 0) sage: T.forward_image(0, 1) # could have equally been (2, 1) (3, 0) """ i = self._bot_labels_to_index[i] if x < self._ring.zero() or x > self._bot_lengths[i]: raise ValueError("x = {} is out of the interval".format(x)) j, y = self._forward_images[i] if x + y < self._top_lengths[j]: return (self._top_labels[j], x + y) x -= self._top_lengths[j] - y j += 1 while x > self._top_lengths[j]: x -= self._top_lengths[j] j += 1 return (self._top_labels[j], x)
[docs] def backward_image(self, i, x): r""" EXAMPLES:: sage: from flatsurf.geometry.interval_exchange_transformation import \ ....: FlowPolygonMap sage: x = polygen(ZZ) sage: K.<sqrt2> = NumberField(x^2 - 2, embedding=AA(2).sqrt()) sage: T = FlowPolygonMap(K, [0,1,2], ....: [sqrt2,1+sqrt2,1], [2,0,1], [1,sqrt2,1+sqrt2]) sage: T.backward_image(*T.forward_image(1, 1)) (1, 1) sage: T.backward_image(*T.forward_image(0, 1)) (0, 1) sage: T.backward_image(*T.forward_image(0, sqrt2-1)) (0, sqrt2 - 1) sage: T.backward_image(*T.forward_image(1, sqrt2-1)) (1, sqrt2 - 1) Singularities are always sent to a ``(i,0)``:: sage: T = FlowPolygonMap(QQ, [2,3,4], [1,1,1], [0,1], [2,1]) sage: T.backward_image(0, 0) (2, 0) sage: T.backward_image(0, 1) # could have equally been (2, 1) (3, 0) TESTS:: sage: T = FlowPolygonMap(K, [0,1,2], ....: [5*sqrt2,1,1], [2,1,0], [1,1,5*sqrt2]) sage: for x in range(1,8): ....: p0 = (0,x) ....: for n in range(5): ....: p1 = T.forward_image(*p0) ....: assert T.backward_image(*p1) == p0, "p0 = {}, p1 = {}".format(p0,p1) ....: p0 = p1 """ i = self._top_labels_to_index[i] if x < self._ring.zero() or x > self._top_lengths[i]: raise ValueError("x = {} is out of the interval".format(x)) j, y = self._backward_images[i] if x + y < self._bot_lengths[j]: return (self._bot_labels[j], x + y) x -= self._bot_lengths[j] - y j += 1 while x > self._bot_lengths[j]: x -= self._bot_lengths[j] j += 1 return (self._bot_labels[j], x)