Ticket #12083: trac12083_tikz_polytope.2.patch

sage/geometry/polyhedra.py

@@ -96 +96 @@
     - Sebastien Barthelemy: documentation improvements, 2008
     - Volker Braun: refactoring, handle non-compact case, 2009 and 2010
     - Andrey Novoseltsev: added Hasse_diagram_from_incidences, 2010
+    - Jean-Philippe Labb\'e: added the tikz method for 2,3-polytope, 2011
 
 TESTS::
 
@@ -119 +120 @@
 
 from subprocess import Popen, PIPE
 from sage.misc.all import tmp_filename
-from sage.misc.functional import norm
+from sage.misc.functional import norm, n
 from sage.misc.package import is_package_installed
 
 from sage.rings.all import Integer, QQ, ZZ, primes_first_n
 from sage.rings.real_double import RDF
 from sage.modules.free_module_element import vector
+from sage.modules.free_module import VectorSpace
 from sage.matrix.constructor import matrix, identity_matrix
 from sage.functions.other import sqrt, floor, ceil
 from sage.functions.trig import sin, cos
+from sage.symbolic.constants import pi
 
 from sage.plot.all import point2d, line2d, arrow, polygon2d
 from sage.plot.plot3d.all import point3d, line3d, arrow3d, polygon3d
+from sage.plot.plot3d.transform import rotate_arbitrary
 from sage.graphs.graph import Graph
 
 from sage.combinat.combinat import permutations
@@ -4875 +4879 @@
         self._restricted_automorphism_group = group
         return group
 
+    def tikz(self, view=[0,0,1], rot_angle=0, scale=2,
+             edge_color='blue!95!black', facet_color='blue!95!black',
+             opacity=0.8, vertex_color='green', axis=False):
+        """
+        Return a string ``tikz_pic`` consisting of a tikz picture of ``self``
+        according to a projection ``view`` and an angle ``rot_angle``
+        obtained via Jmol through the current state property.
+
+        INPUT:
+
+        - ``view`` -- a list of length 3 representing a vector;
+        - ``rot_angle`` -- an angle in degree from 0 to 360;
+        - ``scale`` -- an integer specifying the scaling of the tikz picture;
+        - ``edge_color``, ``facet_color``, ``vertex_color`` -- string representing
+          colors which tikz recognize;
+        - ``opacity`` -- a real number between 0 and 1 giving the opacity of
+          the front facets;
+        - ``axis`` -- a Boolean, to draw the axes at the origin.
+
+        OUTPUT:
+
+        - ``tikz_pic`` -- a string containing the raw text of a TikZ picture.
+
+        .. NOTE::
+
+            The inputs ``view`` and ``rot_angle`` can be obtained from the viewer
+            Jmol:
+
+            1) Right click on the image;
+            2) Select ``Console`;
+            3) Select the tab ``State`;
+            4) Scroll to the line ``moveto``
+
+            It read something like:
+
+            moveto 0.0 {x y z angle} Scale
+
+            The ``view`` is then [x,y,z] and ``rot_angle`` is angle.
+            The following number is the scale.
+
+        EXAMPLES::
+
+            sage: P1 = polytopes.small_rhombicuboctahedron()
+            sage: Image1 = P1.tikz([1,3,5], 175, scale=4)
+            sage: open('polytope-tikz.tex', 'w').write(Image1)
+
+            sage: P2 = Polyhedron(vertices=[[1, 1],[1, 2],[2, 1]])
+            sage: Image2 = P2.tikz(scale=3, edge_color='blue!95!black', facet_color='orange!95!black', opacity=0.4, vertex_color='yellow', axis=True)
+            sage: open('polytope-tikz2.tex', 'w').write(Image2)
+
+            sage: P3 = Polyhedron(vertices=[[-1, -1, 2],[-1, 2, -1],[2, -1, -1]])
+            sage: P3
+            A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 3 vertices.
+            sage: Image3 = P3.tikz([0.5,-1,-0.1], 55, scale=3, edge_color='blue!95!black', facet_color='orange!95!black', opacity=0.7, vertex_color='yellow', axis=True)
+            sage: open('polytope-tikz3.tex', 'w').write(Image3)
+
+            sage: P=Polyhedron(vertices=[[1,1,0,0],[1,2,0,0],[2,1,0,0],[0,0,1,0],[0,0,0,1]])
+            sage: P
+            A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 5 vertices.
+            sage: P.tikz()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: The polytope has to live in 2 or 3 dimensions.
+
+
+        .. TO DO::
+
+            Make it possible to draw Schlegel diagram for 4-polytope.
+
+                sage: P=Polyhedron(vertices=[[1,1,0,0],[1,2,0,0],[2,1,0,0],[0,0,1,0],[0,0,0,1]])
+                sage: P
+                A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 5 vertices.
+                sage: P.tikz()
+                Traceback (most recent call last):
+                ...
+                NotImplementedError: The polytope has to live in 2 or 3 dimensions.
+
+            Make it possible to draw 3-polytope living in higher dimension.
+
+        """
+        if self.ambient_dim() > 3 or self.ambient_dim() < 2:
+            raise NotImplementedError("The polytope has to live in 2 or 3 dimensions.")
+        elif self.ambient_dim() == 2:
+            return self._tikz_2d(scale, edge_color, facet_color, opacity,
+                                 vertex_color, axis)
+        elif self.dim() == 2:
+            return self._tikz_2d_in_3d(view, rot_angle, scale, edge_color,
+                                       facet_color, opacity, vertex_color, axis)
+        else:
+            return self._tikz_3d_in_3d(view, rot_angle, scale, edge_color,
+                                       facet_color, opacity, vertex_color, axis)
+
+    def _tikz_2d(self, scale, edge_color, facet_color, opacity, vertex_color, axis):
+        """
+        Return a string ``tikz_pic`` consisting of a tikz picture of ``self``
+
+        INPUT:
+
+        - ``scale`` -- an integer specifying the scaling of the tikz picture;
+        - ``edge_color``, ``facet_color``, ``vertex_color`` -- string representing
+          colors which tikz recognize;
+        - ``opacity`` -- a real number between 0 and 1 giving the opacity of
+          the front facets;
+        - ``axis`` -- a Boolean, to draw the axes at the origin.
+
+        .. NOTE::
+
+            The ``facet_color`` is the filing color of the polytope (polygon).
+        """
+
+        # Creates the nodes, coordinate and tag for every vertex of the polytope.
+        # The tag is used to draw the front facets later on.
+        dict_drawing = {}
+        edges = ''
+        for v in self.Vrep_generator():
+            v_vect = str([i.n(digits=3) for i in v.vector()])
+            v_vect = v_vect.replace('[','(')
+            v_vect = v_vect.replace(']',')')
+            tag = '%s' %v_vect
+            node = "\\node[%s] at %s     {};\n" %('vertex',tag)
+
+            for nei in v.neighbors():
+                nei_vect = str([i.n(digits=3) for i in nei.vector()])
+                nei_vect = nei_vect.replace('[','(')
+                nei_vect = nei_vect.replace(']',')')
+                tag_nei = '%s' %nei_vect
+                edges += "\\draw[%s] %s -- %s;\n" %('edge',tag,tag_nei)
+
+            coord = '\coordinate %s at %s;\n' %(tag,tag)
+            dict_drawing[v] = node,coord,tag
+
+        # Start to write the output
+        tikz_pic = ''
+        tikz_pic += '\\begin{tikzpicture}%\n'
+        tikz_pic += '\t[scale=%f,\n' %scale
+        tikz_pic += '\tback/.style={loosely dotted, thin},\n'
+        tikz_pic += '\tedge/.style={color=%s, thick},\n' %edge_color
+        tikz_pic += '\tfacet/.style={fill=%s,fill opacity=%f},\n' %(facet_color,opacity)
+        tikz_pic += '\tvertex/.style={inner sep=1pt,circle,draw=%s!25!black,' %vertex_color
+        tikz_pic += 'fill=%s!75!black,thick,anchor=base}]\n\n\n' %vertex_color
+
+        # Draws the axes if True
+        if axis:
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};\n'
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};\n'
+
+        # Create the coordinate of the vertices:
+        tikz_pic += '%% Coordinate of the vertices:\n\n'
+        for v in self.Vrep_generator():
+            tikz_pic += dict_drawing[v][1]
+
+        # Draw the interior by going in a cycle
+        tikz_pic += '\n\n%% Drawing the interior\n\n'
+        cyclic_vert = cyclic_sort_vertices_2d(list(self.Vrep_generator()))
+        tikz_pic += '\\fill[facet] '
+        for v in cyclic_vert:
+            tikz_pic += '%s -- ' %dict_drawing[v][2]
+        tikz_pic += 'cycle {};\n'
+
+        # Draw the edges in the front
+        tikz_pic += '\n\n%% Drawing edges in the front\n\n'
+        tikz_pic += edges
+
+        # Finally, the vertices in front are drawn on top of everything.
+        tikz_pic += '\n\n%% Drawing the vertices in the front\n\n'
+        for v in self.Vrep_generator():
+            tikz_pic += dict_drawing[v][0]
+        tikz_pic += '\n\n\\end{tikzpicture}'
+
+        return tikz_pic
+
+    def _tikz_2d_in_3d(self, view, rot_angle, scale, edge_color, facet_color,
+                       opacity, vertex_color, axis):
+        """
+        Return a string ``tikz_pic`` consisting of a tikz picture of ``self``
+        according to a projection ``view`` and an angle ``rot_angle``
+        obtained via Jmol through the current state property.
+
+        INPUT:
+
+        - ``view`` -- a list of length 3 representing a vector;
+        - ``rot_angle`` -- an angle in degree from 0 to 360 (rotation along the view);
+        - ``scale`` -- an integer specifying the scaling of the tikz picture;
+        - ``edge_color``, ``facet_color``, ``vertex_color`` -- string
+          representing colors which tikz recognize;
+        - ``opacity`` -- a real number between 0 and 1 giving the opacity of
+          the front facets;
+        - ``axis`` -- a Boolean, to draw the axes at the origin.
+
+        .. NOTE::
+
+            The ``facet_color`` is the filing color of the polytope (polygon).
+
+        """
+
+        V = VectorSpace(RDF,3)
+        view_vector = V(view)
+        rot = rotate_arbitrary(view_vector,-(rot_angle/360)*2*pi)
+        rotation_matrix = rot[:2].transpose()
+
+        # Creates the nodes, coordinate and tag for every vertex of the polytope.
+        # The tag is used to draw the front facets later on.
+        dict_drawing = {}
+        edges = ''
+        for v in self.Vrep_generator():
+            v_vect = str([i.n(digits=3) for i in v.vector()])
+            v_vect = v_vect.replace('[','(')
+            v_vect = v_vect.replace(']',')')
+            tag = '%s' %v_vect
+            node = "\\node[%s] at %s     {};\n" %('vertex',tag)
+
+            for nei in v.neighbors():
+                nei_vect = str([i.n(digits=3) for i in nei.vector()])
+                nei_vect = nei_vect.replace('[','(')
+                nei_vect = nei_vect.replace(']',')')
+                tag_nei = '%s' %nei_vect
+                edges += "\\draw[%s] %s -- %s;\n" %('edge',tag,tag_nei)
+
+            coord = '\coordinate %s at %s;\n' %(tag,tag)
+            dict_drawing[v] = node,coord,tag
+
+        # Start to write the output
+        tikz_pic = ''
+        tikz_pic += '\\begin{tikzpicture}%\n'
+        tikz_pic += '\t[x={(%fcm, %fcm)},\n' %(n(rotation_matrix[0][0]),n(rotation_matrix[0][1]))
+        tikz_pic += '\ty={(%fcm, %fcm)},\n' %(n(rotation_matrix[1][0]),n(rotation_matrix[1][1]))
+        tikz_pic += '\tz={(%fcm, %fcm)},\n' %(n(rotation_matrix[2][0]),n(rotation_matrix[2][1]))
+        tikz_pic += '\tscale=%f,\n' %scale
+        tikz_pic += '\tback/.style={loosely dotted, thin},\n'
+        tikz_pic += '\tedge/.style={color=%s, thick},\n' %edge_color
+        tikz_pic += '\tfacet/.style={fill=%s,fill opacity=%f},\n' %(facet_color,opacity)
+        tikz_pic += '\tvertex/.style={inner sep=1pt,circle,draw=%s!25!black,' %vertex_color
+        tikz_pic += 'fill=%s!75!black,thick,anchor=base}]\n\n\n' %vertex_color
+
+        # Draws the axes if True
+        if axis:
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};\n'
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};\n'
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};\n'
+
+        # Create the coordinate of the vertices:
+        tikz_pic += '%% Coordinate of the vertices:\n\n'
+        for v in self.Vrep_generator():
+            tikz_pic += dict_drawing[v][1]
+
+        # Draw the interior by going in a cycle
+        tikz_pic += '\n\n%% Drawing the interior\n\n'
+        cyclic_vert = cyclic_sort_vertices_2d(list(self.Vrep_generator()))
+        tikz_pic += '\\fill[facet] '
+        for v in cyclic_vert:
+            tikz_pic += '%s -- ' %dict_drawing[v][2]
+        tikz_pic += 'cycle {};\n'
+
+        # Draw the edges in the front
+        tikz_pic += '\n\n%% Drawing edges in the front\n\n'
+        tikz_pic += edges
+
+        # Finally, the vertices in front are drawn on top of everything.
+        tikz_pic += '\n\n%% Drawing the vertices in the front\n\n'
+        for v in self.Vrep_generator():
+            tikz_pic += dict_drawing[v][0]
+        tikz_pic += '\n\n\\end{tikzpicture}'
+
+        return tikz_pic
+
+    def _tikz_3d_in_3d(self, view, rot_angle, scale, edge_color,
+                       facet_color, opacity, vertex_color, axis):
+        """
+        Return a string ``tikz_pic`` consisting of a tikz picture of ``self``
+        according to a projection ``view`` and an angle ``rot_angle``
+        obtained via Jmol through the current state property.
+
+        INPUT:
+
+        - ``view`` -- a list of length 3 representing a vector;
+        - ``rot_angle`` -- an angle in degree from 0 to 360;
+        - ``scale`` -- an integer specifying the scaling of the tikz picture;
+        - ``edge_color``, ``facet_color``, ``vertex_color`` -- string
+          representing colors which tikz recognize;
+        - ``opacity`` -- a real number between 0 and 1 giving the opacity of
+          the front facets;
+        - ``axis`` -- a Boolean, to draw the axes at the origin.
+
+        """
+        V = VectorSpace(RDF,3)
+        view_vector = V(view)
+        rot = rotate_arbitrary(view_vector,-(rot_angle/360)*2*pi)
+        rotation_matrix = rot[:2].transpose()
+        proj_vector = (rot**(-1))*V([0,0,1])
+
+        # First compute the back and front vertices and facets
+        front_facets = []
+        back_facets = []
+        for facet in self.Hrep_generator():
+            A = facet.vector()[1:]
+            B = facet.vector()[0]
+            if A*(2000*proj_vector)+B<0:
+                front_facets += [facet]
+            else:
+                back_facets += [facet]
+
+        front_vertices = []
+        for facet in front_facets:
+            A = facet.vector()[1:]
+            B = facet.vector()[0]
+            for v in self.Vrep_generator():
+                if A*v.vector()+B<0.0005:
+                    front_vertices += [v]
+
+        back_vertices = []
+        for facet in back_facets:
+            A = facet.vector()[1:]
+            B = facet.vector()[0]
+            for v in self.Vrep_generator():
+                if A*v.vector()+B<0.0005:
+                    back_vertices += [v]
+
+        # Creates the nodes, coordinate and tag for every vertex of the polytope.
+        # The tag is used to draw the front facets later on.
+        dict_drawing = {}
+        first_part = ''
+        second_part = ''
+        for v in self.Vrep_generator():
+            v_vect = str([i.n(digits=3) for i in v.vector()])
+            v_vect = v_vect.replace('[','(')
+            v_vect = v_vect.replace(']',')')
+            tag = '%s' %v_vect
+            node = "\\node[%s] at %s     {};\n" %('vertex',tag)
+            for nei in v.neighbors():
+                nei_vect = str([i.n(digits=3) for i in nei.vector()])
+                nei_vect = nei_vect.replace('[','(')
+                nei_vect = nei_vect.replace(']',')')
+                tag_nei = '%s' %nei_vect
+                H_v = set(v.incident())
+                H_nei = set(nei.incident())
+                H_v_nei = [h in back_facets for h in H_v.intersection(H_nei)]
+
+                # The back edge has to be between two vertices in the Back
+                # AND such that the 2 facets touching them are in the Back
+                if v in back_vertices and nei in back_vertices and sum(H_v_nei)==2:
+                    first_part += "\\draw[%s,back] %s -- %s;\n" %('edge',tag,tag_nei)
+                else:
+                    second_part += "\\draw[%s] %s -- %s;\n" %('edge',tag,tag_nei)
+            coord = '\coordinate %s at %s;\n' %(tag,tag)
+            dict_drawing[v] = node,coord,tag
+
+        # Start to write the output
+        tikz_pic = ''
+        tikz_pic += '\\begin{tikzpicture}%\n'
+        tikz_pic += '\t[x={(%fcm, %fcm)},\n' %(n(rotation_matrix[0][0]),n(rotation_matrix[0][1]))
+        tikz_pic += '\ty={(%fcm, %fcm)},\n' %(n(rotation_matrix[1][0]),n(rotation_matrix[1][1]))
+        tikz_pic += '\tz={(%fcm, %fcm)},\n' %(n(rotation_matrix[2][0]),n(rotation_matrix[2][1]))
+        tikz_pic += '\tscale=%f,\n' %scale
+        tikz_pic += '\tback/.style={loosely dotted, thin},\n'
+        tikz_pic += '\tedge/.style={color=%s, thick},\n' %edge_color
+        tikz_pic += '\tfacet/.style={fill=%s,fill opacity=%f},\n' %(facet_color,opacity)
+        tikz_pic += '\tvertex/.style={inner sep=1pt,circle,draw=%s!25!black,' %vertex_color
+        tikz_pic += 'fill=%s!75!black,thick,anchor=base}]\n\n\n' %vertex_color
+
+        # Draws the axes if True
+        if axis:
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};\n'
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};\n'
+            tikz_pic += '\\draw[color=black,thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};\n'
+
+        # Create the coordinate of the vertices
+        tikz_pic += '%% Coordinate of the vertices:\n\n'
+        for v in self.Vrep_generator():
+            tikz_pic += dict_drawing[v][1]
+
+        # Draw the edges in the back
+        tikz_pic += '\n\n%% Drawing edges in the back\n\n'
+        tikz_pic += first_part
+
+        # Draw the vertices on top of the back-edges
+        tikz_pic += '\n\n%% Drawing vertices in the back\n\n'
+        for v in self.Vrep_generator():
+            if v in back_vertices and v not in front_vertices:
+                tikz_pic += dict_drawing[v][0]
+
+        # Draw the facets in the front by going in cycles for every facet.
+        tikz_pic += '\n\n%% Drawing the facets\n\n'
+        for facet in front_facets:
+            cyclic_vert = cyclic_sort_vertices_2d(list(facet.incident()))
+            tikz_pic += '\\fill[facet] '
+            for v in cyclic_vert:
+                tikz_pic += '%s -- ' %dict_drawing[v][2]
+            tikz_pic += 'cycle {};\n'
+
+        # Draw the edges in the front
+        tikz_pic += '\n\n%% Drawing edges in the front\n\n'
+        tikz_pic += second_part
+
+        # Finally, the vertices in front are drawn on top of everything.
+        tikz_pic += '\n\n%% Drawing the vertices in the front\n\n'
+        for v in self.Vrep_generator():
+            if v in front_vertices:
+                tikz_pic += dict_drawing[v][0]
+        tikz_pic += '\n\n\\end{tikzpicture}'
+
+        return tikz_pic
 
 #############################################################
 def cyclic_sort_vertices_2d(Vlist):