def __init__(self, projection_direction, height=1.1):
"""
Initializes the projection.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: proj = ProjectionFuncSchlegel([2,2,2])
sage: proj.__init__([2,2,2])
sage: proj(vector([1.1,1.1,1.11]))[0]
0.0302...
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_dir = vector(RDF, projection_direction)
if norm(self.projection_dir).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
self.dim = self.projection_dir.degree()
spcenter = height * self.projection_dir / norm(self.projection_dir)
self.height = height
v = vector(RDF, [0.0] * (self.dim - 1) + [self.height]) - spcenter
polediff = matrix(RDF, v).transpose()
denom = (polediff.transpose() * polediff)[0][0]
if denom.is_zero():
self.house = identity_matrix(RDF, self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom #Householder reflector
def __init__(self, projection_direction, height = 1.1):
"""
Initializes the projection.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: proj = ProjectionFuncSchlegel([2,2,2])
sage: proj.__init__([2,2,2])
sage: proj(vector([1.1,1.1,1.11]))[0]
0.0302...
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_dir = vector(RDF, projection_direction)
if norm(self.projection_dir).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
self.dim = self.projection_dir.degree()
spcenter = height * self.projection_dir/norm(self.projection_dir)
self.height = height
v = vector(RDF, [0.0]*(self.dim-1) + [self.height]) - spcenter
polediff = matrix(RDF,v).transpose()
denom = (polediff.transpose()*polediff)[0][0]
if denom.is_zero():
self.house = identity_matrix(RDF,self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom #Householder reflector
def _normalize(x):
r"""
make x of length 1
"""
from sage.misc.functional import norm
x=vector(RR, x)
if norm(x) == 0:
return x
return vector(x/norm(x))
def _normalize(x):
r"""
make x of length 1
"""
from sage.misc.functional import norm
x = vector(x)
if norm(x) == 0:
return x
return vector(x / norm(x))
def orthonormal_1(dim_n=5):
"""
A matrix of rational approximations to orthonormal vectors to
``(1,...,1)``.
INPUT:
- ``dim_n`` - the dimension of the vectors
OUTPUT:
A matrix over ``QQ`` whose rows are close to an orthonormal
basis to the subspace normal to ``(1,...,1)``.
EXAMPLES::
sage: from sage.geometry.polyhedron.library import Polytopes
sage: m = Polytopes.orthonormal_1(5)
sage: m
[ 70711/100000 -7071/10000 0 0 0]
[ 1633/4000 1633/4000 -81649/100000 0 0]
[ 7217/25000 7217/25000 7217/25000 -43301/50000 0]
[ 22361/100000 22361/100000 22361/100000 22361/100000 -44721/50000]
"""
pb = []
for i in range(0,dim_n-1):
pb.append([1.0/(i+1)]*(i+1) + [-1] + [0]*(dim_n-i-2))
m = matrix(RDF,pb)
new_m = []
for i in range(0,dim_n-1):
new_m.append([RDF(100000*q/norm(m[i])).ceil()/100000 for q in m[i]])
return matrix(QQ,new_m)
def _arc(p,q,s,**kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p,q,s = map( lambda x: vector(x), [p,q,s])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
if abs(det(matrix([p-s,q-s])))<0.01:
return line((p,q),**kwds)
(cx,cy)=var('cx','cy')
equations=[
2*cx*(s[0]-p[0])+2*cy*(s[1]-p[1]) == s[0]**2+s[1]**2-p[0]**2-p[1]**2,
2*cx*(s[0]-q[0])+2*cy*(s[1]-q[1]) == s[0]**2+s[1]**2-q[0]**2-q[1]**2
]
c = vector( [solve( equations, (cx,cy), solution_dict=True )[0][i] for i in [cx,cy]] )
r = norm(p-c)
a_p,a_q,a_s = map( _to_angle, [p-c,q-c,s-c])
angles = [a_p,a_q,a_s]
angles.sort()
if a_s == angles[0]:
return arc( c, r, angle=angles[2], sector=(0,2*pi-angles[2]+angles[1]), **kwds)
if a_s == angles[1]:
return arc( c, r, angle=angles[0], sector=(0,angles[2]-angles[0]), **kwds)
if a_s == angles[2]:
return arc( c, r, angle=angles[1], sector=(0,2*pi-angles[1]+angles[0]), **kwds)
def __init__(self, projection_point):
"""
Create a stereographic projection function.
INPUT:
- ``projection_point`` -- a list of coordinates in the
appropriate dimension, which is the point projected from.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
sage: proj = ProjectionFuncStereographic([1.0,1.0])
sage: proj.__init__([1.0,1.0])
sage: proj.house
[-0.7071067811... 0.7071067811...]
[ 0.7071067811... 0.7071067811...]
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_point = vector(projection_point)
self.dim = self.projection_point.degree()
pproj = vector(RDF, self.projection_point)
self.psize = norm(pproj)
if (self.psize).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
v = vector(RDF, [0.0] * (self.dim - 1) + [self.psize]) - pproj
polediff = matrix(RDF, v).transpose()
denom = RDF((polediff.transpose() * polediff)[0][0])
if denom.is_zero():
self.house = identity_matrix(RDF, self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom # Householder reflector
def orthonormal_1(dim_n=5):
"""
A matrix of rational approximations to orthonormal vectors to
``(1,...,1)``.
INPUT:
- ``dim_n`` - the dimension of the vectors
OUTPUT:
A matrix over ``QQ`` whose rows are close to an orthonormal
basis to the subspace normal to ``(1,...,1)``.
EXAMPLES::
sage: from sage.geometry.polyhedron.library import Polytopes
sage: m = Polytopes.orthonormal_1(5)
sage: m
[ 70711/100000 -7071/10000 0 0 0]
[ 1633/4000 1633/4000 -81649/100000 0 0]
[ 7217/25000 7217/25000 7217/25000 -43301/50000 0]
[ 22361/100000 22361/100000 22361/100000 22361/100000 -44721/50000]
"""
pb = []
for i in range(0, dim_n - 1):
pb.append([1.0 / (i + 1)] * (i + 1) + [-1] + [0] * (dim_n - i - 2))
m = matrix(RDF, pb)
new_m = []
for i in range(0, dim_n - 1):
new_m.append(
[RDF(100000 * q / norm(m[i])).ceil() / 100000 for q in m[i]])
return matrix(QQ, new_m)
def __init__(self, projection_point):
"""
Create a stereographic projection function.
INPUT:
- ``projection_point`` -- a list of coordinates in the
appropriate dimension, which is the point projected from.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
sage: proj = ProjectionFuncStereographic([1.0,1.0])
sage: proj.__init__([1.0,1.0])
sage: proj.house
[-0.7071067811... 0.7071067811...]
[ 0.7071067811... 0.7071067811...]
sage: TestSuite(proj).run(skip='_test_pickling')
"""
self.projection_point = vector(projection_point)
self.dim = self.projection_point.degree()
pproj = vector(RDF,self.projection_point)
self.psize = norm(pproj)
if (self.psize).is_zero():
raise ValueError, "projection direction must be a non-zero vector."
v = vector(RDF, [0.0]*(self.dim-1) + [self.psize]) - pproj
polediff = matrix(RDF,v).transpose()
denom = RDF((polediff.transpose()*polediff)[0][0])
if denom.is_zero():
self.house = identity_matrix(RDF,self.dim)
else:
self.house = identity_matrix(RDF,self.dim) \
- 2*polediff*polediff.transpose()/denom # Householder reflector
def _arc(p, q, s, **kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p, q, s = map(lambda x: vector(x), [p, q, s])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
if abs(det(matrix([p - s, q - s]))) < 0.01:
return line((p, q), **kwds)
(cx, cy) = var('cx', 'cy')
equations = [
2 * cx * (s[0] - p[0]) + 2 * cy * (s[1] - p[1]) == s[0]**2 + s[1]**2 -
p[0]**2 - p[1]**2, 2 * cx * (s[0] - q[0]) + 2 * cy *
(s[1] - q[1]) == s[0]**2 + s[1]**2 - q[0]**2 - q[1]**2
]
c = vector([
solve(equations, (cx, cy), solution_dict=True)[0][i] for i in [cx, cy]
])
r = norm(p - c)
a_p, a_q, a_s = map(lambda x: atan2(x[1], x[0]), [p - c, q - c, s - c])
a_p, a_q = sorted([a_p, a_q])
if a_s < a_p or a_s > a_q:
return arc(c, r, angle=a_q, sector=(0, 2 * pi - a_q + a_p), **kwds)
return arc(c, r, angle=a_p, sector=(0, a_q - a_p), **kwds)
def _stereo_coordinates(x, north=(1,0,0), right=(0,1,0), translation=-1):
r"""
Project stereographically points from a sphere
"""
from sage.misc.functional import norm
north=_normalize(north)
right=vector(right)
right=_normalize(right-(right*north)*north)
if norm(right) == 0:
raise ValueError ("Right must not be linearly dependent from north")
top=north.cross_product(right)
x=_normalize(x)
p=(translation-north*x)/(1-north*x)*(north-x)+x
return vector((right*p, top*p ))
def _stereo_coordinates(x, north=(1, 0, 0), right=(0, 1, 0), translation=-1):
r"""
Project stereographically points from a sphere
"""
from sage.misc.functional import norm
north = _normalize(north)
right = vector(right)
right = _normalize(right - (right * north) * north)
if norm(right) == 0:
raise ValueError("Right must not be linearly dependent from north")
top = north.cross_product(right)
x = _normalize(x)
p = (translation - north * x) / (1 - north * x) * (north - x) + x
return vector((right * p, top * p))
def __call__(self, x):
"""
Apply the projection to a vector.
- ``x`` -- a vector or anything convertible to a vector.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: proj = ProjectionFuncSchlegel([2,2,2])
sage: proj.__call__([1,2,3])
(0.56162854..., 2.09602626...)
"""
v = vector(RDF, x)
if v.is_zero():
raise ValueError, "The origin must not be a vertex."
v = v / norm(v) # normalize vertices to unit sphere
v = self.house * v # reflect so self.projection_dir is at "north pole"
denom = self.height - v[self.dim - 1]
if denom.is_zero():
raise ValueError, 'Point cannot coincide with ' \
'coordinate singularity at ' + repr(x)
return vector(RDF, [v[i] / denom for i in range(self.dim - 1)])
def __call__(self, x):
"""
Apply the projection to a vector.
- ``x`` -- a vector or anything convertible to a vector.
EXAMPLES::
sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: proj = ProjectionFuncSchlegel([2,2,2])
sage: proj.__call__([1,2,3])
(0.56162854..., 2.09602626...)
"""
v = vector(RDF,x)
if v.is_zero():
raise ValueError, "The origin must not be a vertex."
v = v/norm(v) # normalize vertices to unit sphere
v = self.house*v # reflect so self.projection_dir is at "north pole"
denom = self.height-v[self.dim-1]
if denom.is_zero():
raise ValueError, 'Point cannot coincide with ' \
'coordinate singularity at ' + repr(x)
return vector(RDF, [ v[i]/denom for i in range(self.dim-1) ])
# FIXME: refactor this before publishing
r"""
Usese parametric_plot3d() to plot arcs of a circle.
We only plot arcs spanning algles strictly less than pi.
"""
# For sanity purposes convert the input to vectors
from sage.misc.functional import norm
from sage.modules.free_module_element import vector
from sage.functions.trig import arccos, cos, sin
from sage.plot.plot3d.parametric_plot3d import parametric_plot3d
center=vector(center)
first_point=vector(first_point)
second_point=vector(second_point)
first_vector=first_point-center
second_vector=second_point-center
radius=norm(first_vector)
if norm(second_vector)!=radius:
raise ValueError("Ellipse not implemented")
first_unit_vector=first_vector/radius
second_unit_vector=second_vector/radius
normal_vector=second_vector-(second_vector*first_unit_vector)*first_unit_vector
if norm(normal_vector)==0:
print (first_point,second_point)
return
normal_unit_vector=normal_vector/norm(normal_vector)
scalar_product=first_unit_vector*second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle=arccos(scalar_product)
var('t')
return parametric_plot3d(center+first_vector*cos(t)+radius*normal_unit_vector*sin(t),(0,angle),**kwds)
# FIXME: refactor this before publishing
r"""
Usese parametric_plot3d() to plot arcs of a circle.
We only plot arcs spanning algles strictly less than pi.
"""
# For sanity purposes convert the input to vectors
from sage.misc.functional import norm
from sage.modules.free_module_element import vector
from sage.functions.trig import arccos, cos, sin
from sage.plot.plot3d.parametric_plot3d import parametric_plot3d
center = vector(center)
first_point = vector(first_point)
second_point = vector(second_point)
first_vector = first_point - center
second_vector = second_point - center
radius = norm(first_vector)
if norm(second_vector) != radius:
raise ValueError("Ellipse not implemented")
first_unit_vector = first_vector / radius
second_unit_vector = second_vector / radius
normal_vector = second_vector - (second_vector *
first_unit_vector) * first_unit_vector
if norm(normal_vector) == 0:
print(first_point, second_point)
return
normal_unit_vector = normal_vector / norm(normal_vector)
scalar_product = first_unit_vector * second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle = arccos(scalar_product)
var('t')
