Indexed face sets¶

Graphics3D object that consists of a list of polygons, also used for triangulations of other objects.

Usually these objects are not created directly by users.

AUTHORS:

Robert Bradshaw (2007-08-26): initial version
Robert Bradshaw (2007-08-28): significant optimizations

Todo

Smooth triangles using vertex normals

class sage.plot.plot3d.index_face_set.EdgeIter[source]¶

Bases: object

A class for iteration over edges

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: len(list(S.edges())) == 12   # indirect doctest
True

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> len(list(S.edges())) == Integer(12)   # indirect doctest
True

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
len(list(S.edges())) == 12   # indirect doctest

class sage.plot.plot3d.index_face_set.FaceIter[source]¶

Bases: object

A class for iteration over faces

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: len(list(S.faces())) == 6   # indirect doctest
True

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> len(list(S.faces())) == Integer(6)   # indirect doctest
True

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
len(list(S.faces())) == 6   # indirect doctest

class sage.plot.plot3d.index_face_set.IndexFaceSet[source]¶

Bases: PrimitiveObject

Graphics3D object that consists of a list of polygons, also used for triangulations of other objects.

Polygons (mostly triangles and quadrilaterals) are stored in the c struct face_c (see transform.pyx). Rather than storing the points directly for each polygon, each face consists a list of pointers into a common list of points which are basically triples of doubles in a point_c.

Moreover, each face has an attribute color which is used to store color information when faces are colored. The red/green/blue components are then available as floats between 0 and 1 using color.r,color.g,color.b.

Usually these objects are not created directly by users.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: S = IndexFaceSet([[(1,0,0),(0,1,0),(0,0,1)], [(1,0,0),(0,1,0),(0,0,0)]])
sage: S.face_list()
[[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 0.0)]]
sage: S.vertex_list()
[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0), (0.0, 0.0, 0.0)]

sage: def make_face(n): return [(0,0,n),(0,1,n),(1,1,n),(1,0,n)]
sage: S = IndexFaceSet([make_face(n) for n in range(10)])
sage: S.show()

sage: point_list = [(1,0,0),(0,1,0)] + [(0,0,n) for n in range(10)]
sage: face_list = [[0,1,n] for n in range(2,10)]
sage: S = IndexFaceSet(face_list, point_list, color='red')
sage: S.face_list()
[[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 0.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 2.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 3.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 4.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 5.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 6.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 7.0)]]
sage: S.show()

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> S = IndexFaceSet([[(Integer(1),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(1))], [(Integer(1),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(0))]])
>>> S.face_list()
[[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 0.0)]]
>>> S.vertex_list()
[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0), (0.0, 0.0, 0.0)]

>>> def make_face(n): return [(Integer(0),Integer(0),n),(Integer(0),Integer(1),n),(Integer(1),Integer(1),n),(Integer(1),Integer(0),n)]
>>> S = IndexFaceSet([make_face(n) for n in range(Integer(10))])
>>> S.show()

>>> point_list = [(Integer(1),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0))] + [(Integer(0),Integer(0),n) for n in range(Integer(10))]
>>> face_list = [[Integer(0),Integer(1),n] for n in range(Integer(2),Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, color='red')
>>> S.face_list()
[[(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 0.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 1.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 2.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 3.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 4.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 5.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 6.0)],
 [(1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, 0.0, 7.0)]]
>>> S.show()

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
S = IndexFaceSet([[(1,0,0),(0,1,0),(0,0,1)], [(1,0,0),(0,1,0),(0,0,0)]])
S.face_list()
S.vertex_list()
def make_face(n): return [(0,0,n),(0,1,n),(1,1,n),(1,0,n)]
S = IndexFaceSet([make_face(n) for n in range(10)])
S.show()
point_list = [(1,0,0),(0,1,0)] + [(0,0,n) for n in range(10)]
face_list = [[0,1,n] for n in range(2,10)]
S = IndexFaceSet(face_list, point_list, color='red')
S.face_list()
S.show()

A simple example of colored IndexFaceSet (Issue #12212):

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: from sage.plot.plot3d.texture import Texture
sage: point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
sage: face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
sage: col = rainbow(10, 'rgbtuple')
sage: t_list = [Texture(col[i]) for i in range(10)]
sage: S = IndexFaceSet(face_list, point_list, texture_list=t_list)
sage: S.show(viewer='tachyon')

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> from sage.plot.plot3d.texture import Texture
>>> point_list = [(Integer(2),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(1),Integer(1)),(Integer(1),Integer(0),Integer(1)),(Integer(1),Integer(1),Integer(0))]
>>> face_list = [[Integer(0),Integer(4),Integer(5)],[Integer(3),Integer(4),Integer(5)],[Integer(2),Integer(3),Integer(4)],[Integer(1),Integer(3),Integer(5)]]
>>> col = rainbow(Integer(10), 'rgbtuple')
>>> t_list = [Texture(col[i]) for i in range(Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, texture_list=t_list)
>>> S.show(viewer='tachyon')

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
from sage.plot.plot3d.texture import Texture
point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
col = rainbow(10, 'rgbtuple')
t_list = [Texture(col[i]) for i in range(10)]
S = IndexFaceSet(face_list, point_list, texture_list=t_list)
S.show(viewer='tachyon')

add_condition(condition, N=100, eps=1e-06)[source]¶

Cut the surface according to the given condition.

This allows to take the intersection of the surface with a domain in 3-space, in such a way that the result has a smooth boundary.

INPUT:

condition – boolean function on ambient space, that defines the domain
N – max number of steps used by the bisection method (default: 100) to cut the boundary triangles that are not entirely within the domain.
eps – target accuracy in the intersection (default: 1.0e-6)

OUTPUT: an IndexFaceSet

This will contain both triangular and quadrilateral faces.

EXAMPLES:

Sage

sage: var('x,y,z')                                                          # needs sage.symbolic
(x, y, z)
sage: P = implicit_plot3d(z-x*y,(-2,2),(-2,2),(-2,2))                       # needs sage.symbolic
sage: def condi(x, y, z):
....:     return bool(x*x+y*y+z*z <= Integer(1))
sage: R = P.add_condition(condi, 20); R                                     # needs sage.symbolic
Graphics3d Object

Python

>>> from sage.all import *
>>> var('x,y,z')                                                          # needs sage.symbolic
(x, y, z)
>>> P = implicit_plot3d(z-x*y,(-Integer(2),Integer(2)),(-Integer(2),Integer(2)),(-Integer(2),Integer(2)))                       # needs sage.symbolic
>>> def condi(x, y, z):
...     return bool(x*x+y*y+z*z <= Integer(Integer(1)))
>>> R = P.add_condition(condi, Integer(20)); R                                     # needs sage.symbolic
Graphics3d Object

Sage Live

var('x,y,z')                                                          # needs sage.symbolic
P = implicit_plot3d(z-x*y,(-2,2),(-2,2),(-2,2))                       # needs sage.symbolic
def condi(x, y, z):
    return bool(x*x+y*y+z*z <= Integer(1))
R = P.add_condition(condi, 20); R                                     # needs sage.symbolic

An example with colors:

Sage

sage: def condi(x, y, z):
....:     return bool(x*x+y*y <= 1.1)
sage: cm = colormaps.hsv
sage: cf = lambda x,y,z: float(x+y) % 1
sage: P = implicit_plot3d(x**2+y**2+z**2-1-x**2*z+y**2*z,                   # needs sage.symbolic
....:                     (-2,2),(-2,2),(-2,2),color=(cm,cf))
sage: R = P.add_condition(condi,40); R                                      # needs sage.symbolic
Graphics3d Object

Python

>>> from sage.all import *
>>> def condi(x, y, z):
...     return bool(x*x+y*y <= RealNumber('1.1'))
>>> cm = colormaps.hsv
>>> cf = lambda x,y,z: float(x+y) % Integer(1)
>>> P = implicit_plot3d(x**Integer(2)+y**Integer(2)+z**Integer(2)-Integer(1)-x**Integer(2)*z+y**Integer(2)*z,                   # needs sage.symbolic
...                     (-Integer(2),Integer(2)),(-Integer(2),Integer(2)),(-Integer(2),Integer(2)),color=(cm,cf))
>>> R = P.add_condition(condi,Integer(40)); R                                      # needs sage.symbolic
Graphics3d Object

Sage Live

def condi(x, y, z):
    return bool(x*x+y*y <= 1.1)
cm = colormaps.hsv
cf = lambda x,y,z: float(x+y) % 1
P = implicit_plot3d(x**2+y**2+z**2-1-x**2*z+y**2*z,                   # needs sage.symbolic
                    (-2,2),(-2,2),(-2,2),color=(cm,cf))
R = P.add_condition(condi,40); R                                      # needs sage.symbolic

An example with transparency:

Sage

sage: P = implicit_plot3d(x**4+y**4+z**2-4, (x,-2,2), (y,-2,2), (z,-2,2),   # needs sage.symbolic
....:                     alpha=0.3)
sage: def cut(a, b, c):
....:     return a*a+c*c > 2
sage: Q = P.add_condition(cut,40); Q                                        # needs sage.symbolic
Graphics3d Object

Python

>>> from sage.all import *
>>> P = implicit_plot3d(x**Integer(4)+y**Integer(4)+z**Integer(2)-Integer(4), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), (z,-Integer(2),Integer(2)),   # needs sage.symbolic
...                     alpha=RealNumber('0.3'))
>>> def cut(a, b, c):
...     return a*a+c*c > Integer(2)
>>> Q = P.add_condition(cut,Integer(40)); Q                                        # needs sage.symbolic
Graphics3d Object

Sage Live

P = implicit_plot3d(x**4+y**4+z**2-4, (x,-2,2), (y,-2,2), (z,-2,2),   # needs sage.symbolic
                    alpha=0.3)
def cut(a, b, c):
    return a*a+c*c > 2
Q = P.add_condition(cut,40); Q                                        # needs sage.symbolic

A sombrero with quadrilaterals:

Sage

sage: P = plot3d(-sin(2*x*x+2*y*y)*exp(-x*x-y*y), (x,-2,2), (y,-2,2),       # needs sage.symbolic
....:            color='gold')
sage: def cut(x, y, z):
....:     return x*x+y*y < 1
sage: Q = P.add_condition(cut);Q                                            # needs sage.symbolic
Graphics3d Object

Python

>>> from sage.all import *
>>> P = plot3d(-sin(Integer(2)*x*x+Integer(2)*y*y)*exp(-x*x-y*y), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)),       # needs sage.symbolic
...            color='gold')
>>> def cut(x, y, z):
...     return x*x+y*y < Integer(1)
>>> Q = P.add_condition(cut);Q                                            # needs sage.symbolic
Graphics3d Object

Sage Live

P = plot3d(-sin(2*x*x+2*y*y)*exp(-x*x-y*y), (x,-2,2), (y,-2,2),       # needs sage.symbolic
           color='gold')
def cut(x, y, z):
    return x*x+y*y < 1
Q = P.add_condition(cut);Q                                            # needs sage.symbolic

Todo

Use a dichotomy to search for the place where to cut,
Compute the cut only once for each edge.

bounding_box()[source]¶

Calculate the bounding box for the vertices in this object (ignoring infinite or NaN coordinates).

OUTPUT:

a tuple ( (low_x, low_y, low_z), (high_x, high_y, high_z)), which gives the coordinates of opposite corners of the bounding box.

EXAMPLES:

Sage

sage: x,y = var('x,y')                                                      # needs sage.symbolic
sage: p = plot3d(sqrt(sin(x)*sin(y)), (x,0,2*pi), (y,0,2*pi))               # needs sage.symbolic
sage: p.bounding_box()                                                      # needs sage.symbolic
((0.0, 0.0, 0.0), (6.283185307179586, 6.283185307179586, 0.9991889981715697))

Python

>>> from sage.all import *
>>> x,y = var('x,y')                                                      # needs sage.symbolic
>>> p = plot3d(sqrt(sin(x)*sin(y)), (x,Integer(0),Integer(2)*pi), (y,Integer(0),Integer(2)*pi))               # needs sage.symbolic
>>> p.bounding_box()                                                      # needs sage.symbolic
((0.0, 0.0, 0.0), (6.283185307179586, 6.283185307179586, 0.9991889981715697))

Sage Live

x,y = var('x,y')                                                      # needs sage.symbolic
p = plot3d(sqrt(sin(x)*sin(y)), (x,0,2*pi), (y,0,2*pi))               # needs sage.symbolic
p.bounding_box()                                                      # needs sage.symbolic

dual(**kwds)[source]¶

Return the dual.

EXAMPLES:

Sage

sage: S = cube()
sage: T = S.dual()
sage: len(T.vertex_list())
6

Python

>>> from sage.all import *
>>> S = cube()
>>> T = S.dual()
>>> len(T.vertex_list())
6

Sage Live

S = cube()
T = S.dual()
len(T.vertex_list())

edge_list()[source]¶

Return the list of edges.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: S.edge_list()[0]
((1.0, -2.0, 3.0), (1.0, 2.0, 3.0))

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> S.edge_list()[Integer(0)]
((1.0, -2.0, 3.0), (1.0, 2.0, 3.0))

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
S.edge_list()[0]

edges()[source]¶

An iterator over the edges.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: list(S.edges())[0]
((1.0, -2.0, 3.0), (1.0, 2.0, 3.0))

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> list(S.edges())[Integer(0)]
((1.0, -2.0, 3.0), (1.0, 2.0, 3.0))

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
list(S.edges())[0]

face_list(render_params=None)[source]¶

Return the list of faces.

Every face is given as a tuple of vertices.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: S.face_list(S.default_render_params())[0]
[(1.0, 2.0, 3.0), (-1.0, 2.0, 3.0), (-1.0, -2.0, 3.0), (1.0, -2.0, 3.0)]

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> S.face_list(S.default_render_params())[Integer(0)]
[(1.0, 2.0, 3.0), (-1.0, 2.0, 3.0), (-1.0, -2.0, 3.0), (1.0, -2.0, 3.0)]

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
S.face_list(S.default_render_params())[0]

faces()[source]¶

An iterator over the faces.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: list(S.faces()) == S.face_list()
True

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> list(S.faces()) == S.face_list()
True

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
list(S.faces()) == S.face_list()

has_local_colors()[source]¶

Return True if and only if every face has an individual color.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: from sage.plot.plot3d.texture import Texture
sage: point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
sage: face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
sage: col = rainbow(10, 'rgbtuple')
sage: t_list=[Texture(col[i]) for i in range(10)]
sage: S = IndexFaceSet(face_list, point_list, texture_list=t_list)
sage: S.has_local_colors()
True

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: S.has_local_colors()
False

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> from sage.plot.plot3d.texture import Texture
>>> point_list = [(Integer(2),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(1),Integer(1)),(Integer(1),Integer(0),Integer(1)),(Integer(1),Integer(1),Integer(0))]
>>> face_list = [[Integer(0),Integer(4),Integer(5)],[Integer(3),Integer(4),Integer(5)],[Integer(2),Integer(3),Integer(4)],[Integer(1),Integer(3),Integer(5)]]
>>> col = rainbow(Integer(10), 'rgbtuple')
>>> t_list=[Texture(col[i]) for i in range(Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, texture_list=t_list)
>>> S.has_local_colors()
True

>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> S.has_local_colors()
False

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
from sage.plot.plot3d.texture import Texture
point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
col = rainbow(10, 'rgbtuple')
t_list=[Texture(col[i]) for i in range(10)]
S = IndexFaceSet(face_list, point_list, texture_list=t_list)
S.has_local_colors()
from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
S.has_local_colors()

index_faces()[source]¶

Return the list over all faces of the indices of the vertices.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: S.index_faces()
[[0, 1, 2, 3],
 [0, 4, 5, 1],
 [0, 3, 6, 4],
 [5, 4, 6, 7],
 [6, 3, 2, 7],
 [2, 1, 5, 7]]

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> S.index_faces()
[[0, 1, 2, 3],
 [0, 4, 5, 1],
 [0, 3, 6, 4],
 [5, 4, 6, 7],
 [6, 3, 2, 7],
 [2, 1, 5, 7]]

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
S.index_faces()

index_faces_with_colors()[source]¶

Return the list over all faces of (indices of the vertices, color).

This only works if every face has its own color.

See also

has_local_colors()

EXAMPLES:

A simple colored one:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: from sage.plot.plot3d.texture import Texture
sage: point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
sage: face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
sage: col = rainbow(10, 'rgbtuple')
sage: t_list = [Texture(col[i]) for i in range(10)]
sage: S = IndexFaceSet(face_list, point_list, texture_list=t_list)
sage: S.index_faces_with_colors()
[([0, 4, 5], '#ff0000'),
 ([3, 4, 5], '#ff9900'),
 ([2, 3, 4], '#cbff00'),
 ([1, 3, 5], '#33ff00')]

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> from sage.plot.plot3d.texture import Texture
>>> point_list = [(Integer(2),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(1),Integer(1)),(Integer(1),Integer(0),Integer(1)),(Integer(1),Integer(1),Integer(0))]
>>> face_list = [[Integer(0),Integer(4),Integer(5)],[Integer(3),Integer(4),Integer(5)],[Integer(2),Integer(3),Integer(4)],[Integer(1),Integer(3),Integer(5)]]
>>> col = rainbow(Integer(10), 'rgbtuple')
>>> t_list = [Texture(col[i]) for i in range(Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, texture_list=t_list)
>>> S.index_faces_with_colors()
[([0, 4, 5], '#ff0000'),
 ([3, 4, 5], '#ff9900'),
 ([2, 3, 4], '#cbff00'),
 ([1, 3, 5], '#33ff00')]

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
from sage.plot.plot3d.texture import Texture
point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
col = rainbow(10, 'rgbtuple')
t_list = [Texture(col[i]) for i in range(10)]
S = IndexFaceSet(face_list, point_list, texture_list=t_list)
S.index_faces_with_colors()

When the texture is global, an error is raised:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: S.index_faces_with_colors()
Traceback (most recent call last):
...
ValueError: the texture is global

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> S.index_faces_with_colors()
Traceback (most recent call last):
...
ValueError: the texture is global

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
S.index_faces_with_colors()

is_enclosed()[source]¶

Whether or not it is necessary to render the back sides of the polygons.

One is assuming, of course, that they have the correct orientation.

This is may be passed in on construction. It is also calculated in sage.plot.plot3d.parametric_surface.ParametricSurface by verifying the opposite edges of the rendered domain either line up or are pinched together.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: IndexFaceSet([[(0,0,1),(0,1,0),(1,0,0)]]).is_enclosed()
False

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> IndexFaceSet([[(Integer(0),Integer(0),Integer(1)),(Integer(0),Integer(1),Integer(0)),(Integer(1),Integer(0),Integer(0))]]).is_enclosed()
False

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
IndexFaceSet([[(0,0,1),(0,1,0),(1,0,0)]]).is_enclosed()

jmol_repr(render_params)[source]¶: Return a jmol representation for self.

json_repr(render_params)[source]¶: Return a json representation for self.

obj_repr(render_params)[source]¶: Return an obj representation for self.

partition(f)[source]¶

Partition the faces of self.

The partition is done according to the value of a map \(f: \RR^3 \rightarrow \ZZ\) applied to the center of each face.

INPUT:

f – a function from \(\RR^3\) to \(\ZZ\)

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: len(S.partition(lambda x,y,z: floor(x+y+z)))
6

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> len(S.partition(lambda x,y,z: floor(x+y+z)))
6

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
len(S.partition(lambda x,y,z: floor(x+y+z)))

sticker(face_list, width, hover, **kwds)[source]¶: Return a sticker on the chosen faces.

stickers(colors, width, hover)[source]¶

Return a group of IndexFaceSets.

INPUT:

colors – list of colors/textures to use (in cyclic order)
width – offset perpendicular into the edge (to create a border) may also be negative
hover – offset normal to the face (usually have to float above the original surface so it shows, typically this value is very small compared to the actual object

OUTPUT: Graphics3dGroup of stickers

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import Box
sage: B = Box(.5,.4,.3, color='black')
sage: S = B.stickers(['red','yellow','blue'], 0.1, 0.05)
sage: S.show()
sage: (S+B).show()

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import Box
>>> B = Box(RealNumber('.5'),RealNumber('.4'),RealNumber('.3'), color='black')
>>> S = B.stickers(['red','yellow','blue'], RealNumber('0.1'), RealNumber('0.05'))
>>> S.show()
>>> (S+B).show()

Sage Live

from sage.plot.plot3d.shapes import Box
B = Box(.5,.4,.3, color='black')
S = B.stickers(['red','yellow','blue'], 0.1, 0.05)
S.show()
(S+B).show()

stl_binary_repr(render_params)[source]¶

Return data for STL (STereoLithography) representation of the surface.

The STL binary representation is a list of binary strings, one for each triangle.

EXAMPLES:

Sage

sage: G = sphere()
sage: data = G.stl_binary_repr(G.default_render_params()); len(data)
1368

Python

>>> from sage.all import *
>>> G = sphere()
>>> data = G.stl_binary_repr(G.default_render_params()); len(data)
1368

Sage Live

G = sphere()
data = G.stl_binary_repr(G.default_render_params()); len(data)

tachyon_repr(render_params)[source]¶

Return a tachyon object for self.

EXAMPLES:

A basic test with a triangle:

Sage

sage: G = polygon([(0,0,1), (1,1,1), (2,0,1)])
sage: s = G.tachyon_repr(G.default_render_params()); s
['TRI V0 0 0 1 V1 1 1 1 V2 2 0 1', ...]

Python

>>> from sage.all import *
>>> G = polygon([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1)), (Integer(2),Integer(0),Integer(1))])
>>> s = G.tachyon_repr(G.default_render_params()); s
['TRI V0 0 0 1 V1 1 1 1 V2 2 0 1', ...]

Sage Live

G = polygon([(0,0,1), (1,1,1), (2,0,1)])
s = G.tachyon_repr(G.default_render_params()); s

A simple colored one:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: from sage.plot.plot3d.texture import Texture
sage: point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
sage: face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
sage: col = rainbow(10, 'rgbtuple')
sage: t_list = [Texture(col[i]) for i in range(10)]
sage: S = IndexFaceSet(face_list, point_list, texture_list=t_list)
sage: S.tachyon_repr(S.default_render_params())
['TRI V0 2 0 0 V1 1 0 1 V2 1 1 0',
'TEXTURE... AMBIENT 0.3 DIFFUSE 0.7 SPECULAR 0 OPACITY 1.0... COLOR 1 0 0 ... TEXFUNC 0',...]

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> from sage.plot.plot3d.texture import Texture
>>> point_list = [(Integer(2),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(1),Integer(1)),(Integer(1),Integer(0),Integer(1)),(Integer(1),Integer(1),Integer(0))]
>>> face_list = [[Integer(0),Integer(4),Integer(5)],[Integer(3),Integer(4),Integer(5)],[Integer(2),Integer(3),Integer(4)],[Integer(1),Integer(3),Integer(5)]]
>>> col = rainbow(Integer(10), 'rgbtuple')
>>> t_list = [Texture(col[i]) for i in range(Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, texture_list=t_list)
>>> S.tachyon_repr(S.default_render_params())
['TRI V0 2 0 0 V1 1 0 1 V2 1 1 0',
'TEXTURE... AMBIENT 0.3 DIFFUSE 0.7 SPECULAR 0 OPACITY 1.0... COLOR 1 0 0 ... TEXFUNC 0',...]

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
from sage.plot.plot3d.texture import Texture
point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
col = rainbow(10, 'rgbtuple')
t_list = [Texture(col[i]) for i in range(10)]
S = IndexFaceSet(face_list, point_list, texture_list=t_list)
S.tachyon_repr(S.default_render_params())

threejs_repr(render_params)[source]¶

Return representation of the surface suitable for plotting with three.js.

EXAMPLES:

A simple triangle:

Sage

sage: G = polygon([(0,0,1), (1,1,1), (2,0,1)])
sage: G.threejs_repr(G.default_render_params())
[('surface',
  {'color': '#0000ff',
   'faces': [[0, 1, 2]],
   'opacity': 1.0,
   'vertices': [{'x': 0.0, 'y': 0.0, 'z': 1.0},
    {'x': 1.0, 'y': 1.0, 'z': 1.0},
    {'x': 2.0, 'y': 0.0, 'z': 1.0}]})]

Python

>>> from sage.all import *
>>> G = polygon([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1)), (Integer(2),Integer(0),Integer(1))])
>>> G.threejs_repr(G.default_render_params())
[('surface',
  {'color': '#0000ff',
   'faces': [[0, 1, 2]],
   'opacity': 1.0,
   'vertices': [{'x': 0.0, 'y': 0.0, 'z': 1.0},
    {'x': 1.0, 'y': 1.0, 'z': 1.0},
    {'x': 2.0, 'y': 0.0, 'z': 1.0}]})]

Sage Live

G = polygon([(0,0,1), (1,1,1), (2,0,1)])
G.threejs_repr(G.default_render_params())

The same but with more options applied:

Sage

sage: G = polygon([(0,0,1), (1,1,1), (2,0,1)], color='red', opacity=0.5,
....:             render_order=2, threejs_flat_shading=True,
....:             single_side=True, mesh=True, thickness=10, depth_write=True)
sage: G.threejs_repr(G.default_render_params())
[('surface',
  {'color': '#ff0000',
   'depthWrite': True,
   'faces': [[0, 1, 2]],
   'linewidth': 10.0,
   'opacity': 0.5,
   'renderOrder': 2.0,
   'showMeshGrid': True,
   'singleSide': True,
   'useFlatShading': True,
   'vertices': [{'x': 0.0, 'y': 0.0, 'z': 1.0},
    {'x': 1.0, 'y': 1.0, 'z': 1.0},
    {'x': 2.0, 'y': 0.0, 'z': 1.0}]})]

Python

>>> from sage.all import *
>>> G = polygon([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1)), (Integer(2),Integer(0),Integer(1))], color='red', opacity=RealNumber('0.5'),
...             render_order=Integer(2), threejs_flat_shading=True,
...             single_side=True, mesh=True, thickness=Integer(10), depth_write=True)
>>> G.threejs_repr(G.default_render_params())
[('surface',
  {'color': '#ff0000',
   'depthWrite': True,
   'faces': [[0, 1, 2]],
   'linewidth': 10.0,
   'opacity': 0.5,
   'renderOrder': 2.0,
   'showMeshGrid': True,
   'singleSide': True,
   'useFlatShading': True,
   'vertices': [{'x': 0.0, 'y': 0.0, 'z': 1.0},
    {'x': 1.0, 'y': 1.0, 'z': 1.0},
    {'x': 2.0, 'y': 0.0, 'z': 1.0}]})]

Sage Live

G = polygon([(0,0,1), (1,1,1), (2,0,1)], color='red', opacity=0.5,
            render_order=2, threejs_flat_shading=True,
            single_side=True, mesh=True, thickness=10, depth_write=True)
G.threejs_repr(G.default_render_params())

vertex_list()[source]¶

Return the list of vertices.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = polygon([(0,0,1), (1,1,1), (2,0,1)])
sage: S.vertex_list()[0]
(0.0, 0.0, 1.0)

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = polygon([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1)), (Integer(2),Integer(0),Integer(1))])
>>> S.vertex_list()[Integer(0)]
(0.0, 0.0, 1.0)

Sage Live

from sage.plot.plot3d.shapes import *
S = polygon([(0,0,1), (1,1,1), (2,0,1)])
S.vertex_list()[0]

vertices()[source]¶

An iterator over the vertices.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Cone(1,1)
sage: list(S.vertices()) == S.vertex_list()
True

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Cone(Integer(1),Integer(1))
>>> list(S.vertices()) == S.vertex_list()
True

Sage Live

from sage.plot.plot3d.shapes import *
S = Cone(1,1)
list(S.vertices()) == S.vertex_list()

x3d_geometry()[source]¶

Return the x3d data.

EXAMPLES:

A basic test with a triangle:

Sage

sage: G = polygon([(0,0,1), (1,1,1), (2,0,1)])
sage: print(G.x3d_geometry())

<IndexedFaceSet coordIndex='0,1,2,-1'>
  <Coordinate point='0.0 0.0 1.0,1.0 1.0 1.0,2.0 0.0 1.0'/>
</IndexedFaceSet>

Python

>>> from sage.all import *
>>> G = polygon([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1)), (Integer(2),Integer(0),Integer(1))])
>>> print(G.x3d_geometry())
<BLANKLINE>
<IndexedFaceSet coordIndex='0,1,2,-1'>
  <Coordinate point='0.0 0.0 1.0,1.0 1.0 1.0,2.0 0.0 1.0'/>
</IndexedFaceSet>
<BLANKLINE>

Sage Live

G = polygon([(0,0,1), (1,1,1), (2,0,1)])
print(G.x3d_geometry())

A simple colored one:

Sage

sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: from sage.plot.plot3d.texture import Texture
sage: point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
sage: face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
sage: col = rainbow(10, 'rgbtuple')
sage: t_list = [Texture(col[i]) for i in range(10)]
sage: S = IndexFaceSet(face_list, point_list, texture_list=t_list)
sage: print(S.x3d_geometry())

<IndexedFaceSet solid='False' colorPerVertex='False' coordIndex='0,4,5,-1,3,4,5,-1,2,3,4,-1,1,3,5,-1'>
  <Coordinate point='2.0 0.0 0.0,0.0 2.0 0.0,0.0 0.0 2.0,0.0 1.0 1.0,1.0 0.0 1.0,1.0 1.0 0.0'/>
  <Color color='1.0 0.0 0.0,1.0 0.6000000000000001 0.0,0.7999999999999998 1.0 0.0,0.20000000000000018 1.0 0.0' />
</IndexedFaceSet>

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import IndexFaceSet
>>> from sage.plot.plot3d.texture import Texture
>>> point_list = [(Integer(2),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(1),Integer(1)),(Integer(1),Integer(0),Integer(1)),(Integer(1),Integer(1),Integer(0))]
>>> face_list = [[Integer(0),Integer(4),Integer(5)],[Integer(3),Integer(4),Integer(5)],[Integer(2),Integer(3),Integer(4)],[Integer(1),Integer(3),Integer(5)]]
>>> col = rainbow(Integer(10), 'rgbtuple')
>>> t_list = [Texture(col[i]) for i in range(Integer(10))]
>>> S = IndexFaceSet(face_list, point_list, texture_list=t_list)
>>> print(S.x3d_geometry())
<BLANKLINE>
<IndexedFaceSet solid='False' colorPerVertex='False' coordIndex='0,4,5,-1,3,4,5,-1,2,3,4,-1,1,3,5,-1'>
  <Coordinate point='2.0 0.0 0.0,0.0 2.0 0.0,0.0 0.0 2.0,0.0 1.0 1.0,1.0 0.0 1.0,1.0 1.0 0.0'/>
  <Color color='1.0 0.0 0.0,1.0 0.6000000000000001 0.0,0.7999999999999998 1.0 0.0,0.20000000000000018 1.0 0.0' />
</IndexedFaceSet>
<BLANKLINE>

Sage Live

from sage.plot.plot3d.index_face_set import IndexFaceSet
from sage.plot.plot3d.texture import Texture
point_list = [(2,0,0),(0,2,0),(0,0,2),(0,1,1),(1,0,1),(1,1,0)]
face_list = [[0,4,5],[3,4,5],[2,3,4],[1,3,5]]
col = rainbow(10, 'rgbtuple')
t_list = [Texture(col[i]) for i in range(10)]
S = IndexFaceSet(face_list, point_list, texture_list=t_list)
print(S.x3d_geometry())

class sage.plot.plot3d.index_face_set.VertexIter[source]¶

Bases: object

A class for iteration over vertices

EXAMPLES:

Sage

sage: from sage.plot.plot3d.shapes import *
sage: S = Box(1,2,3)
sage: len(list(S.vertices())) == 8   # indirect doctest
True

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.shapes import *
>>> S = Box(Integer(1),Integer(2),Integer(3))
>>> len(list(S.vertices())) == Integer(8)   # indirect doctest
True

Sage Live

from sage.plot.plot3d.shapes import *
S = Box(1,2,3)
len(list(S.vertices())) == 8   # indirect doctest

sage.plot.plot3d.index_face_set.cut_edge_by_bisection(pointa, pointb, condition, eps=1e-06, N=100)[source]¶

Cut an intersecting edge using the bisection method.

Given two points (pointa and pointb) and a condition (boolean function), this calculates the position at the edge (defined by both points) where the boolean condition switches its value.

INPUT:

pointa, pointb – two points in 3-dimensional space
N – max number of steps in the bisection method (default: 100) to cut the boundary triangles that are not entirely within the domain.
eps – target accuracy in the intersection (default: 1.0e-6)

OUTPUT:

intersection of the edge defined by pointa and pointb, and condition.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.index_face_set import cut_edge_by_bisection
sage: cut_edge_by_bisection((0.0,0.0,0.0), (1.0,1.0,0.0),
....:                       lambda x,y,z: x**2+y**2+z**2 < 1, eps=1.0E-12)
(0.7071067811864395, 0.7071067811864395, 0.0)

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import cut_edge_by_bisection
>>> cut_edge_by_bisection((RealNumber('0.0'),RealNumber('0.0'),RealNumber('0.0')), (RealNumber('1.0'),RealNumber('1.0'),RealNumber('0.0')),
...                       lambda x,y,z: x**Integer(2)+y**Integer(2)+z**Integer(2) < Integer(1), eps=RealNumber('1.0E-12'))
(0.7071067811864395, 0.7071067811864395, 0.0)

Sage Live

from sage.plot.plot3d.index_face_set import cut_edge_by_bisection
cut_edge_by_bisection((0.0,0.0,0.0), (1.0,1.0,0.0),
                      lambda x,y,z: x**2+y**2+z**2 < 1, eps=1.0E-12)

sage.plot.plot3d.index_face_set.midpoint(pointa, pointb, w)[source]¶

Return the weighted mean of two points in 3-space.

INPUT:

pointa, pointb – two points in 3-dimensional space
w – a real weight between 0 and 1

If the weight is zero, the result is pointb. If the weight is one, the result is pointa.

EXAMPLES:

Sage

sage: from sage.plot.plot3d.index_face_set import midpoint
sage: midpoint((1,2,3),(4,4,4),0.8)
(1.60000000000000, 2.40000000000000, 3.20000000000000)

Python

>>> from sage.all import *
>>> from sage.plot.plot3d.index_face_set import midpoint
>>> midpoint((Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(4),Integer(4)),RealNumber('0.8'))
(1.60000000000000, 2.40000000000000, 3.20000000000000)

Sage Live

from sage.plot.plot3d.index_face_set import midpoint
midpoint((1,2,3),(4,4,4),0.8)

sage.plot.plot3d.index_face_set.sticker(face, width, hover)[source]¶: Return a sticker over the given face.