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Trac #22081: Add new option to hyperbolic graphics to select the model
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As a first approach add the poincare disc model, and an option to select
the model to use the drawing hyperbolic arcs and polygons

URL: https://trac.sagemath.org/22081
Reported by: jhonrubia6
Ticket author(s): Javier Honrubia González
Reviewer(s): Travis Scrimshaw
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Release Manager committed Feb 27, 2022
2 parents 72b70cb + 29f335c commit 8c5f78c
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Showing 6 changed files with 745 additions and 218 deletions.
7 changes: 7 additions & 0 deletions src/doc/en/reference/references/index.rst
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Expand Up @@ -1384,6 +1384,10 @@ REFERENCES:
.. [CFI1992] Cai, JY., Fürer, M. & Immerman, N. Combinatorica (1992) 12: 389.
:doi:`10.1007/BF01305232`
.. [CFKP1997] James W. Cannon, William J. Floyd, Richard Kenyon and Walter R. Parry.
*Hyperbolic Geometry*. Flavors of Geometry, MSRI Publications,
Volume 31, 1997.
.. [CFKPR2010] Ioannis Caragiannis, Afonso Ferreira, Christos Kaklamanis,
Stéphane Pérennes, Hervé Rivano.
*Fractional Path Coloring in Bounded Degree Trees with
Expand Down Expand Up @@ -5308,6 +5312,9 @@ REFERENCES:
Variable I. Lecture Notes in Mathematics,
vol 320. (1973), Springer, Berlin, Heidelberg
.. [Sta1993] Stahl, Saul: *The Poincaré Half-plane: A Gateway to Modern Geometry*
Jones & Bartlett Learning, 1993. ISBN 086720298X, 9780867202984
.. [Sta1995] \R. P. Stanley, *A symmetric function generalization
of the chromatic polynomial of a graph*, Adv. Math., ***111***
no.1 (1995), 166-194. :doi:`10.1006/aima.1995.1020`.
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102 changes: 81 additions & 21 deletions src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
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Expand Up @@ -89,7 +89,7 @@
from sage.rings.cc import CC
from sage.rings.real_mpfr import RR
from sage.plot.arc import arc
from sage.plot.line import line
from sage.plot.bezier_path import bezier_path
from sage.symbolic.constants import pi
from sage.modules.free_module_element import vector
from sage.matrix.constructor import matrix
Expand Down Expand Up @@ -696,15 +696,27 @@ def complete(self):
::
sage: KM = H.KM()
sage: KM.get_geodesic((0,0), (0, 1/2)).complete()
sage: KM.get_geodesic((0,0), (0,1/2)).complete()
Geodesic in KM from (0, -1) to (0, 1)
.. PLOT::
g = HyperbolicPlane().KM().get_geodesic((0.0,0.0), (0.0, 0.5))
g = HyperbolicPlane().KM().get_geodesic(CC(0,0), CC(0, 0.5))
h = g.complete()
sphinx_plot(g.plot()+h.plot(linestyle='dashed'))
::
sage: KM.get_geodesic(-I, 1).complete()
Geodesic in KM from -I to 1
.. PLOT::
g = HyperbolicPlane().KM().get_geodesic(CC(0,-1), CC(1, 0))
h = g.complete()
sphinx_plot(g.plot()+h.plot(linestyle='dashed'))
::
sage: HM = H.HM()
Expand Down Expand Up @@ -1186,12 +1198,15 @@ def plot(self, boundary=True, **options):
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
pic = line((end_1, end_2), **opts)
else:
end_1 = (real(end_1), imag(end_1))
end_2 = (real(end_2), imag(end_2))
pic = bezier_path([[end_1, end_2]], **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
pic += bd_pic
return pic
else:
center = (bd_1 + bd_2) / 2 # Circle center
Expand All @@ -1213,7 +1228,7 @@ def plot(self, boundary=True, **options):
bd_dict = {'bd_min': midpoint - length, 'bd_max': midpoint +
length}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
pic += bd_pic
return pic

def ideal_endpoints(self):
Expand Down Expand Up @@ -1244,10 +1259,10 @@ def ideal_endpoints(self):
M = self._model
# infinity is the first endpoint, so the other ideal endpoint
# is just the real part of the second coordinate
if start == infinity:
if CC(start).is_infinity():
return [M.get_point(start), M.get_point(x2)]
# Same idea as above
if end == infinity:
if CC(end).is_infinity():
return [M.get_point(x1), M.get_point(end)]
# We could also have a vertical line with two interior points
if x1 == x2:
Expand Down Expand Up @@ -2174,11 +2189,9 @@ class HyperbolicGeodesicPD(HyperbolicGeodesic):
g = PD.get_geodesic(I,-I/2)
h = PD.get_geodesic(-0.5+I*0.5,0.5+I*0.5)
sphinx_plot(g.plot()+h.plot(color='green'))
"""

def plot(self, boundary=True, **options):

r"""
Plot ``self``.
Expand Down Expand Up @@ -2235,7 +2248,7 @@ def plot(self, boundary=True, **options):
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
pic = bezier_path([[(real(end_1), imag(end_1)), (real(end_2), imag(end_2))]], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
Expand Down Expand Up @@ -2276,20 +2289,18 @@ class HyperbolicGeodesicKM(HyperbolicGeodesic):
EXAMPLES::
sage: KM = HyperbolicPlane().KM()
sage: g = KM.get_geodesic(KM.get_point((0.1,0.9)), KM.get_point((-0.1,-0.9)))
sage: g = KM.get_geodesic((0.1,0.9),(-0.1,-0.9))
sage: h = KM.get_geodesic((-0.707106781,-0.707106781),(0.707106781,-0.707106781))
sage: P = g.plot(color='orange')+h.plot(); P # optional - sage.plot
Graphics object consisting of 4 graphics primitives
.. PLOT::
KM = HyperbolicPlane().KM()
g = KM.get_geodesic((0.1,0.9),
(-0.1,-0.9))
h = KM.get_geodesic((-0.707106781,-0.707106781),
(0.707106781,-0.707106781))
g = KM.get_geodesic(CC(0.1,0.9),
CC(-0.1,-0.9))
h = KM.get_geodesic(CC(-0.707106781,-0.707106781),
CC(0.707106781,-0.707106781))
sphinx_plot(g.plot(color='orange')+h.plot())
"""
Expand All @@ -2300,19 +2311,24 @@ def plot(self, boundary=True, **options):
EXAMPLES::
sage: HyperbolicPlane().KM().get_geodesic((0,0), (1,0)).plot() # optional - sage.plot
sage: HyperbolicPlane().KM().get_geodesic(0, 1).plot() # optional - sage.plot
Graphics object consisting of 2 graphics primitives
.. PLOT::
KM = HyperbolicPlane().KM()
sphinx_plot(KM.get_geodesic((0,0), (1,0)).plot())
sphinx_plot(KM.get_geodesic(CC(0,0), CC(1,0)).plot())
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
pic = line([k.coordinates() for k in self.endpoints()], **opts)
def map_pt(pt):
if pt in CC:
return CC(pt)
return CC(*pt)
end_1, end_2 = [map_pt(k.coordinates()) for k in self.endpoints()]
pic = bezier_path([[(real(end_1), imag(end_1)), (real(end_2), imag(end_2))]], **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
Expand All @@ -2334,7 +2350,6 @@ class HyperbolicGeodesicHM(HyperbolicGeodesic):
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_geodesic import *
sage: HM = HyperbolicPlane().HM()
sage: p1 = HM.get_point((4, -4, sqrt(33)))
sage: p2 = HM.get_point((-3,-3,sqrt(19)))
Expand All @@ -2350,6 +2365,51 @@ class HyperbolicGeodesicHM(HyperbolicGeodesic):
sphinx_plot(g.plot(color='blue'))
"""
def _plot_vertices(self, points=75):
r"""
Return ``self`` plotting vertices in `\RR^3`.
Auxiliary function needed to plot polygons.
EXAMPLES::
sage: HM = HyperbolicPlane().HM()
sage: p1 = HM.get_point((4, -4, sqrt(33)))
sage: p2 = HM.get_point((-3,-3,sqrt(19)))
sage: g = HM.get_geodesic(p1, p2)
sage: g._plot_vertices(5)
[(4.0, -4.0, 5.744562...),
(1.363213..., -1.637073..., 2.353372...),
(0.138568..., -0.969980..., 1.400022...),
(-0.942533..., -1.307681..., 1.896945...),
(-3.0, -3.0, 4.358898...)]
"""
from sage.plot.misc import setup_for_eval_on_grid
from sage.arith.srange import xsrange

x = SR.var('x')
v1, u2 = [vector(k.coordinates()) for k in self.endpoints()]
# Lorentzian Gram Shmidt. The original vectors will be
# u1, u2 and the orthogonal ones will be v1, v2. Except
# v1 = u1, and I don't want to declare another variable,
# hence the odd naming convention above.
# We need the Lorentz dot product of v1 and u2.
v1_ldot_u2 = u2[0]*v1[0] + u2[1]*v1[1] - u2[2]*v1[2]
v2 = u2 + v1_ldot_u2 * v1
v2_norm = sqrt(v2[0]**2 + v2[1]**2 - v2[2]**2)
v2 = v2 / v2_norm
v2_ldot_u2 = u2[0]*v2[0] + u2[1]*v2[1] - u2[2]*v2[2]
# Now v1 and v2 are Lorentz orthogonal, and |v1| = -1, |v2|=1
# That is, v1 is unit timelike and v2 is unit spacelike.
# This means that cosh(x)*v1 + sinh(x)*v2 is unit timelike.
hyperbola = tuple(cosh(x)*v1 + sinh(x)*v2)
endtime = arcsinh(v2_ldot_u2)
# mimic the function _parametric_plot3d_curve using a bezier3d instead of a line3d
# this is required in order to be able to plot hyperbolic polygons whithin the plot library
g, ranges = setup_for_eval_on_grid(hyperbola, [(x, 0, endtime)], points)
f_x, f_y, f_z = g
points = [(f_x(u), f_y(u), f_z(u)) for u in xsrange(*ranges[0], include_endpoint=True)]
return points

def plot(self, show_hyperboloid=True, **graphics_options):
r"""
Expand Down
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