From c6434469fb6976c72d179974edce0a6ad2e7faba Mon Sep 17 00:00:00 2001 From: William Blattman Date: Sun, 18 Mar 2012 17:03:07 -0700 Subject: [PATCH 1/3] added code to subdivide faces, not quite finished but looks promising --- geometry.py | 163 ++++++++++++++++++++++++++++++++++++++-------------- 1 file changed, 120 insertions(+), 43 deletions(-) diff --git a/geometry.py b/geometry.py index 0bfb70a..d75f49c 100755 --- a/geometry.py +++ b/geometry.py @@ -104,6 +104,8 @@ class Edge(object): self.vertices = [] self.faces = [] self.edges = [] + self.__edgeVertex = None + self.__subEdges = [] def neighborFace(self, neighborFace): if neighborFace == self.faces[0]: @@ -111,27 +113,37 @@ class Edge(object): else: return self.faces[0] - def edgeVertex(self): + def __getSubEdges(self): + if not self.__subEdges: + self.__subEdges = [Edge(), Edge()] + self.__subEdges[0].vertices = [self.vertices[0], self.edgeVertex] + self.__subEdges[1].vertices = [self.edgeVertex, self.vertices[1]] + return self.__subEdges + subEdges = property(fget=__getSubEdges) + + def __getEdgeVertex(self): ''' Set each edge vertices to be the average of the two neighboring face vertices and its two original end vertices. ''' - - # two neighboring face vertices: - neighborVertices = [face.centroid for face in self.faces] - - vertices = list(self.vertices) - vertices.extend(neighborVertices) - return self.__averageVertices(vertices) + if not self.__edgeVertex: + # two neighboring face vertices: + neighboringFaceVertices = [face.centroid for face in self.faces] + neighboringFaceVertices.extend(self.vertices) + xs = [vertex.x for vertex in neighboringFaceVertices] + ys = [vertex.y for vertex in neighboringFaceVertices] + zs = [vertex.z for vertex in neighboringFaceVertices] + x = sum(xs)/len(xs) + y = sum(ys)/len(ys) + z = sum(zs)/len(zs) + self.__edgeVertex = Vertex(x, y, z) + self.__edgeVertex.edges.extend(self.__subEdges) + return self.__edgeVertex + edgeVertex = property(fget=__getEdgeVertex) def __averageVertices(self, vertices): - xs = [vertex.x for vertex in vertices] - ys = [vertex.y for vertex in vertices] - zs = [vertex.z for vertex in vertices] - x = sum(xs)/len(xs) - y = sum(ys)/len(ys) - z = sum(zs)/len(zs) - return Vertex(x, y, z) + + return class Face(object): ''' @@ -139,25 +151,94 @@ class Face(object): ''' def __init__(self): self.edges = [] + self.__centroid = None + self.__interiorEdges = [] + self.__subFaces = [] def __getCentroid(self): - # gather all vertex coords - faceVertices = self.__getFaceVertices() - xs = [vertex.x for vertex in faceVertices] - ys = [vertex.y for vertex in faceVertices] - zs = [vertex.z for vertex in faceVertices] - - # average each vertex component - x = sum(xs) / len(xs) - y = sum(ys) / len(ys) - z = sum(zs) / len(zs) - - return Vertex(x, y, z) + if not self.__centroid: + # gather all face vertex coords + faceVertices = list(set([vertex for edge in self.edges for vertex in edge.vertices])) + xs = [vertex.x for vertex in faceVertices] + ys = [vertex.y for vertex in faceVertices] + zs = [vertex.z for vertex in faceVertices] + + # average each vertex component + x = sum(xs) / len(xs) + y = sum(ys) / len(ys) + z = sum(zs) / len(zs) + + self.__centroid = Vertex(x, y, z) + return self.__centroid centroid = property(fget=__getCentroid) - def __getFaceVertices(self): - return list([vertex for edge in self.edges for vertex in edge.vertices]) - vertices = property(fget=__getFaceVertices) + def __getSubFaces(self): + self.__setupSubDivisions() + return self.__subFaces + subFaces = property(fget=__getSubFaces) + + def __getInteriorEdges(self): + self.__setupSubDivisions() + return self.__interiorEdges + interiorEdges = property(fget=__getInteriorEdges ) + + def __setupSubDivisions(self): + ''' + v0 ev0 v1 + *------e0-----* + | | | + | | | + ev3 e|11----f5----e|1 ev1 + | | | + | | | + *------e2-----* + v3 ev2 v2 + ''' + if not self.__subFaces: + # create empty subFaces that will be filled with edge references below + # these need to at least exist so the interior edges have something to reference + self.__subFaces = [Face() for edge in self.edges] + + if not self.__interiorEdges: + # set up empty edge objects to be filled below + self.__interiorEdges = [Edge() for edge in self.edges] + + # each interior edge connects the exterior edge vertex (mid-point) to the faceVertex (centroid) + for index in range(len(self.edges)): + prevIndex = (index - 1) % len(self.edges) + nextIndex = (index + 1) % len(self.edges) + + # end vertices are face centroid and currEdge edgeVertex + self.__interiorEdges[index].vertices = [self.edges[index].edgeVertex, + self.centroid] + + # wing edges are the current edge's subEdges (ordered same as vertex order) and the prev and next interior edges + self.__interiorEdges[index].edges = [self.edges[index].subEdges[0], + self.edges[index].subEdges[1], + self.__interiorEdges[prevIndex], + self.__interiorEdges[nextIndex]] + + # edge faces are the new subFaces (current and next faces), the current will be define below + # and the next will be defined on the next iteration (or already defined on the last iteration) + self.__interiorEdges[index].faces = [self.__subFaces[index], + self.__subFaces[nextIndex]] + + # now reference the current edge back into the faces, + # and the edge.subEdges, and the edge.edgeVertex + + # current subFace (same index as current interior edge) + # set its edges to reference the same edges used to setup the interior edge + # order will be pretty important on these steps... + self.__subFaces[index].edges = [self.edges[index].subEdges[0], + self.__interiorEdges[index], + self.__interiorEdges[prevIndex], + self.edges[prevIndex].subEdges[1]] + + # just set one of the vertex edges, the other belongs to another face and will get added when that face is run + self.edges[index].edgeVertex.edges.append(self.__interiorEdges[index]) + + self.edges[index].subEdges[0].faces.append(self.__subFaces[index]) + self.edges[index].subEdges[0].faces.append(self.__subFaces[index]) class Polygon(object): ''' @@ -170,10 +251,8 @@ class Polygon(object): self.edges = edges self.faces = faces - def __getFaceNeighbors(self, testFace): - for face in self.faces: - testFace.isNeighbor(face) - + def __repr__(self): + return str([edge for edge in self.edges]) def catmullClarkRefine(self): ''' For each face, add a face vertex @@ -184,17 +263,16 @@ class Polygon(object): For each original vertex P, take the average F of all n face vertices for faces touching P, and take the average R of all n edge midvertices for edges touching P, where each edge midvertex is the average of its two endvertex vertices. Move each original vertex to the vertex ''' - # this calculates and returns the averaged vertex point - print self.faces[0].edges[0].edgeVertex() + # each face knows how to subdivide and create a set of subfaces, including interior edges and setup their references correctly... <- not completely finished... - # this returns a vertex at the face centroid - print self.faces[0].centroid + for face in self.faces: + for subFace in face.subFaces: + for edge in subFace.edges: + for vertex in edge.vertices: + print vertex - # this will get to the first neighborFace... - # each face knows its faceVertex, and all of its edgeVertices - neighborFace = self.faces[0].edges[0].neighborFace(self.faces[0]) + # plotting these in excel seems to show the correct values (at first glace...) - print neighborFace.centroid # so now what......... @@ -290,7 +368,6 @@ polygon = Polygon(v, e, f) polygon.catmullClarkRefine() - # from numpy import * # import pylab # import mpl_toolkits.mplot3d.axes3d as p3 From a93c047a5db3687687b9cb5a519acd61b7cc20b8 Mon Sep 17 00:00:00 2001 From: William Blattman Date: Sun, 18 Mar 2012 17:52:24 -0700 Subject: [PATCH 2/3] simple plotting to see if it works. --- geometry.py | 207 +++++++++++++++++++++++++++------------------------- 1 file changed, 107 insertions(+), 100 deletions(-) diff --git a/geometry.py b/geometry.py index d75f49c..3890ad9 100755 --- a/geometry.py +++ b/geometry.py @@ -246,13 +246,11 @@ class Polygon(object): it needs to average vertices with all adjoinging faces ''' - def __init__(self, vertices, edges, faces): - self.vertices = vertices - self.edges = edges - self.faces = faces + def __init__(self): + self.vertices = [] + self.edges = [] + self.faces = [] - def __repr__(self): - return str([edge for edge in self.edges]) def catmullClarkRefine(self): ''' For each face, add a face vertex @@ -264,123 +262,132 @@ class Polygon(object): ''' # each face knows how to subdivide and create a set of subfaces, including interior edges and setup their references correctly... <- not completely finished... - + p = Polygon() + edges = [] + vertices = [] + faces = [] for face in self.faces: for subFace in face.subFaces: + faces.append(subFace) for edge in subFace.edges: + edges.append(edge) for vertex in edge.vertices: - print vertex - + vertices.append(vertex) + p.faces = faces + p.vertices = vertices + p.edges = edges # plotting these in excel seems to show the correct values (at first glace...) - - # so now what......... - + # so now what......... # (F + 2R + (n-3) P) / n # # F = average of all face vertices touching P # R = average of all edge vertices touching P # P original point # n = face vertices or edge vertices (should be the same number) - - -v = [] -v.append(Vertex(0.0, 1.0, 0.0)) -v.append(Vertex(1.0, 1.0, 0.0)) -v.append(Vertex(1.0, 0.0, 0.0)) -v.append(Vertex(0.0, 0.0, 0.0)) -v.append(Vertex(0.0, 1.0, 1.0)) -v.append(Vertex(1.0, 1.0, 1.0)) -v.append(Vertex(1.0, 0.0, 1.0)) -v.append(Vertex(0.0, 0.0, 1.0)) + return p -e = [] -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) -e.append(Edge()) +def createPolygon(): + v = [] + v.append(Vertex(0.0, 1.0, 0.0)) + v.append(Vertex(1.0, 1.0, 0.0)) + v.append(Vertex(1.0, 0.0, 0.0)) + v.append(Vertex(0.0, 0.0, 0.0)) + v.append(Vertex(0.0, 1.0, 1.0)) + v.append(Vertex(1.0, 1.0, 1.0)) + v.append(Vertex(1.0, 0.0, 1.0)) + v.append(Vertex(0.0, 0.0, 1.0)) -f = [] -f.append(Face()) -f.append(Face()) -f.append(Face()) -f.append(Face()) -f.append(Face()) -f.append(Face()) + e = [] + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) + e.append(Edge()) -# vertex list -v[0].edges = [e[0], e[3], e[4]] -v[1].edges = [e[0], e[5], e[1]] -v[2].edges = [e[1], e[6], e[2]] -v[3].edges = [e[2], e[7], e[3]] -v[4].edges = [e[4], e[11], e[8]] -v[5].edges = [e[5], e[9], e[8]] -v[6].edges = [e[2], e[9], e[10]] -v[7].edges = [e[7], e[10], e[11]] + f = [] + f.append(Face()) + f.append(Face()) + f.append(Face()) + f.append(Face()) + f.append(Face()) + f.append(Face()) -# face list -f[0].edges = [e[0], e[1], e[2], e[3]] -f[1].edges = [e[1], e[5], e[9], e[6]] -f[2].edges = [e[2], e[6], e[10], e[7]] -f[3].edges = [e[4], e[3], e[7], e[11]] -f[4].edges = [e[8], e[5], e[0], e[4]] -f[5].edges = [e[10], e[9], e[8], e[11]] + # vertex list + v[0].edges = [e[0], e[3], e[4]] + v[1].edges = [e[0], e[5], e[1]] + v[2].edges = [e[1], e[6], e[2]] + v[3].edges = [e[2], e[7], e[3]] + v[4].edges = [e[4], e[11], e[8]] + v[5].edges = [e[5], e[9], e[8]] + v[6].edges = [e[2], e[9], e[10]] + v[7].edges = [e[7], e[10], e[11]] -#winged edges ordered by face, then by vertex reference -e[0].vertices, e[0].faces, e[0].edges = [v[0], v[1]], [f[0], f[4]], [e[3], e[1], e[4], e[5]] -e[1].vertices, e[1].faces, e[1].edges = [v[1], v[2]], [f[0], f[1]], [e[0], e[2], e[5], e[6]] -e[2].vertices, e[2].faces, e[2].edges = [v[2], v[3]], [f[0], f[2]], [e[3], e[7], e[1], e[6]] -e[3].vertices, e[3].faces, e[3].edges = [v[3], v[0]], [f[3], f[0]], [e[4], e[7], e[0], e[2]] -e[4].vertices, e[4].faces, e[4].edges = [v[0], v[4]], [f[3], f[4]], [e[11], e[3], e[0], e[8]] -e[5].vertices, e[5].faces, e[5].edges = [v[5], v[1]], [f[4], f[1]], [e[8], e[0], e[9], e[1]] -e[6].vertices, e[6].faces, e[6].edges = [v[2], v[6]], [f[1], f[2]], [e[1], e[9], e[2], e[10]] -e[7].vertices, e[7].faces, e[7].edges = [v[7], v[3]], [f[3], f[2]], [e[11], e[3], e[10], e[2]] -e[8].vertices, e[8].faces, e[8].edges = [v[4], v[5]], [f[4], f[5]], [e[4], e[5], e[11], e[9]] -e[9].vertices, e[9].faces, e[9].edges = [v[5], v[6]], [f[1], f[5]], [e[5], e[6], e[8], e[10]] -e[10].vertices, e[10].faces, e[10].edges = [v[7], v[6]], [f[2], f[5]], [e[7], e[6], e[11], e[9]] -e[11].vertices, e[11].faces, e[11].edges = [v[4], v[7]], [f[3], f[5]], [e[4], e[7], e[8], e[10]] + # face list + f[0].edges = [e[0], e[1], e[2], e[3]] + f[1].edges = [e[1], e[5], e[9], e[6]] + f[2].edges = [e[2], e[6], e[10], e[7]] + f[3].edges = [e[4], e[3], e[7], e[11]] + f[4].edges = [e[8], e[5], e[0], e[4]] + f[5].edges = [e[10], e[9], e[8], e[11]] -# just to prove to myself that the objects are the same, this is what years of pass by value have done to me... + #winged edges ordered by face, then by vertex reference + e[0].vertices, e[0].faces, e[0].edges = [v[0], v[1]], [f[0], f[4]], [e[3], e[1], e[4], e[5]] + e[1].vertices, e[1].faces, e[1].edges = [v[1], v[2]], [f[0], f[1]], [e[0], e[2], e[5], e[6]] + e[2].vertices, e[2].faces, e[2].edges = [v[2], v[3]], [f[0], f[2]], [e[3], e[7], e[1], e[6]] + e[3].vertices, e[3].faces, e[3].edges = [v[3], v[0]], [f[3], f[0]], [e[4], e[7], e[0], e[2]] + e[4].vertices, e[4].faces, e[4].edges = [v[0], v[4]], [f[3], f[4]], [e[11], e[3], e[0], e[8]] + e[5].vertices, e[5].faces, e[5].edges = [v[5], v[1]], [f[4], f[1]], [e[8], e[0], e[9], e[1]] + e[6].vertices, e[6].faces, e[6].edges = [v[2], v[6]], [f[1], f[2]], [e[1], e[9], e[2], e[10]] + e[7].vertices, e[7].faces, e[7].edges = [v[7], v[3]], [f[3], f[2]], [e[11], e[3], e[10], e[2]] + e[8].vertices, e[8].faces, e[8].edges = [v[4], v[5]], [f[4], f[5]], [e[4], e[5], e[11], e[9]] + e[9].vertices, e[9].faces, e[9].edges = [v[5], v[6]], [f[1], f[5]], [e[5], e[6], e[8], e[10]] + e[10].vertices, e[10].faces, e[10].edges = [v[7], v[6]], [f[2], f[5]], [e[7], e[6], e[11], e[9]] + e[11].vertices, e[11].faces, e[11].edges = [v[4], v[7]], [f[3], f[5]], [e[4], e[7], e[8], e[10]] -# print id(v[0].x) -# print id(e[0].vertices[0].x) -# print id(f[0].edges[0].vertices[0].x) + # just to prove to myself that the objects are the same, this is what years of pass by value have done to me... -# v[0].x = 9 + # print id(v[0].x) + # print id(e[0].vertices[0].x) + # print id(f[0].edges[0].vertices[0].x) -# print id(v[0].x) -# print id(e[0].vertices[0].x) -# print id(f[0].edges[0].vertices[0].x) + # v[0].x = 9 -# print v[0].x -# print e[0].vertices[0].x -# print f[0].edges[0].vertices[0].x + # print id(v[0].x) + # print id(e[0].vertices[0].x) + # print id(f[0].edges[0].vertices[0].x) -polygon = Polygon(v, e, f) -polygon.catmullClarkRefine() + # print v[0].x + # print e[0].vertices[0].x + # print f[0].edges[0].vertices[0].x + p = Polygon() + p.vertices = v + p.edges = e + p.faces = f + return p +polygon = createPolygon() +newPolygon = polygon.catmullClarkRefine() -# from numpy import * -# import pylab -# import mpl_toolkits.mplot3d.axes3d as p3 +from numpy import * +import pylab +import mpl_toolkits.mplot3d.axes3d as p3 -# fig = pylab.figure() -# ax = p3.Axes3D(fig) -# ax.plot_wireframe(xs, ys, zs) -# ax.set_xlabel('X') -# ax.set_ylabel('Y') -# ax.set_zlabel('Z') -# pylab.show() - -# v4 = Vertex(4, 0, 0, 1) -# v5 = Vertex(5, 1, 0, 1) -# v6 = Vertex(6, 0, 1, 1) -# v7 = Vertex(7, 1, 1, 1) +fig = pylab.figure() +ax = p3.Axes3D(fig) +for edge in newPolygon.edges: + xs = [vertex.x for vertex in edge.vertices] + ys = [vertex.y for vertex in edge.vertices] + zs = [vertex.z for vertex in edge.vertices] + ax.plot_wireframe(xs, ys, zs) +ax.set_xlabel('X') +ax.set_ylabel('Y') +ax.set_zlabel('Z') +pylab.show() \ No newline at end of file From 805edfe12f312afeda55f0ceaa4a192742359d54 Mon Sep 17 00:00:00 2001 From: William Blattman Date: Sun, 18 Mar 2012 21:57:44 -0700 Subject: [PATCH 3/3] getting closer, its starting to look like it... --- geometry.py | 90 +++++++++++++++++++++++++++++++++++++++++++---------- 1 file changed, 74 insertions(+), 16 deletions(-) diff --git a/geometry.py b/geometry.py index 3890ad9..3dfcc5b 100755 --- a/geometry.py +++ b/geometry.py @@ -88,15 +88,33 @@ class Vertex(object): ''' A vertex is a position along with other information such as color, normal vector and texture coordinates. ''' - def __init__(self, x, y, z): + def __init__(self, x=0, y=0, z=0): self.x = x self.y = y self.z = z self.edges = [] - + def __repr__(self): return "<%.2f, %.2f, %.2f>" % (self.x, self.y, self.z) - + + def __add__(self, other): + # for now just assume type(other) = Vertex... bad, I know + return Vertex(self.x + other.x, self.y + other.y, self.z + other.z) + + def __radd__(self, other): + return self.__add__(other) + + def __mul__(self, other): + # for now just assume type(other) = int or float + return Vertex(self.x * other, self.y * other, self.z * other) + + def __rmul__(self, other): + return self.__mul__(other) + + def __div__(self, other): + # same assumption as __mult__ + return Vertex(self.x / other, self.y / other, self.z / other) + class Edge(object): ''' ''' @@ -112,6 +130,9 @@ class Edge(object): return self.faces[1] else: return self.faces[0] + def __getMidPoint(self): + return sum(self.vertices, Vertex())/len(self.vertices) + midPoint = property(fget=__getMidPoint) def __getSubEdges(self): if not self.__subEdges: @@ -239,7 +260,7 @@ class Face(object): self.edges[index].subEdges[0].faces.append(self.__subFaces[index]) self.edges[index].subEdges[0].faces.append(self.__subFaces[index]) - + class Polygon(object): ''' Face splitting should happend on the polygon level(?). It doesn't make sense to split just one face since @@ -261,11 +282,17 @@ class Polygon(object): For each original vertex P, take the average F of all n face vertices for faces touching P, and take the average R of all n edge midvertices for edges touching P, where each edge midvertex is the average of its two endvertex vertices. Move each original vertex to the vertex ''' + + + + + # each face knows how to subdivide and create a set of subfaces, including interior edges and setup their references correctly... <- not completely finished... p = Polygon() edges = [] vertices = [] faces = [] + for face in self.faces: for subFace in face.subFaces: faces.append(subFace) @@ -273,9 +300,39 @@ class Polygon(object): edges.append(edge) for vertex in edge.vertices: vertices.append(vertex) + + for vertex in self.vertices: + faceVertices = [] + edgeMidPoints = [] + for edge in vertex.edges: + edgeMidPoints.append(edge.midPoint) + for face in edge.faces: + faceVertices.append(face.centroid) + + f = sum(list(set(faceVertices)), Vertex())/len(list(set(faceVertices))) + + r = sum(list(set(edgeMidPoints)), Vertex())/len(list(set(edgeMidPoints))) + p = vertex + n = len(vertex.edges) + v = (f + (2.0 * r) + (n - 3.0) * p) / n + # print v + vertex.x = v.x + vertex.y = v.y + vertex.z = v.z + + # so now what......... + # (F + 2R + (n-3) P) / n + # + # F = average of all face vertices touching P + # R = average of all edge vertices touching P + # P original point + # n = number of edges connecting to P + p.faces = faces p.vertices = vertices p.edges = edges + + # plotting these in excel seems to show the correct values (at first glace...) # so now what......... @@ -286,13 +343,14 @@ class Polygon(object): # P original point # n = face vertices or edge vertices (should be the same number) return p - + def createPolygon(): v = [] v.append(Vertex(0.0, 1.0, 0.0)) v.append(Vertex(1.0, 1.0, 0.0)) v.append(Vertex(1.0, 0.0, 0.0)) v.append(Vertex(0.0, 0.0, 0.0)) + v.append(Vertex(0.0, 1.0, 1.0)) v.append(Vertex(1.0, 1.0, 1.0)) v.append(Vertex(1.0, 0.0, 1.0)) @@ -327,7 +385,7 @@ def createPolygon(): v[3].edges = [e[2], e[7], e[3]] v[4].edges = [e[4], e[11], e[8]] v[5].edges = [e[5], e[9], e[8]] - v[6].edges = [e[2], e[9], e[10]] + v[6].edges = [e[6], e[9], e[10]] v[7].edges = [e[7], e[10], e[11]] # face list @@ -339,16 +397,16 @@ def createPolygon(): f[5].edges = [e[10], e[9], e[8], e[11]] #winged edges ordered by face, then by vertex reference - e[0].vertices, e[0].faces, e[0].edges = [v[0], v[1]], [f[0], f[4]], [e[3], e[1], e[4], e[5]] - e[1].vertices, e[1].faces, e[1].edges = [v[1], v[2]], [f[0], f[1]], [e[0], e[2], e[5], e[6]] - e[2].vertices, e[2].faces, e[2].edges = [v[2], v[3]], [f[0], f[2]], [e[3], e[7], e[1], e[6]] - e[3].vertices, e[3].faces, e[3].edges = [v[3], v[0]], [f[3], f[0]], [e[4], e[7], e[0], e[2]] - e[4].vertices, e[4].faces, e[4].edges = [v[0], v[4]], [f[3], f[4]], [e[11], e[3], e[0], e[8]] - e[5].vertices, e[5].faces, e[5].edges = [v[5], v[1]], [f[4], f[1]], [e[8], e[0], e[9], e[1]] - e[6].vertices, e[6].faces, e[6].edges = [v[2], v[6]], [f[1], f[2]], [e[1], e[9], e[2], e[10]] - e[7].vertices, e[7].faces, e[7].edges = [v[7], v[3]], [f[3], f[2]], [e[11], e[3], e[10], e[2]] - e[8].vertices, e[8].faces, e[8].edges = [v[4], v[5]], [f[4], f[5]], [e[4], e[5], e[11], e[9]] - e[9].vertices, e[9].faces, e[9].edges = [v[5], v[6]], [f[1], f[5]], [e[5], e[6], e[8], e[10]] + e[0].vertices, e[0].faces, e[0].edges = [v[0], v[1]], [f[0], f[4]], [e[3], e[1], e[4], e[5]] + e[1].vertices, e[1].faces, e[1].edges = [v[1], v[2]], [f[0], f[1]], [e[0], e[2], e[5], e[6]] + e[2].vertices, e[2].faces, e[2].edges = [v[2], v[3]], [f[0], f[2]], [e[1], e[3], e[6], e[7]] + e[3].vertices, e[3].faces, e[3].edges = [v[3], v[0]], [f[3], f[0]], [e[4], e[7], e[0], e[2]] + e[4].vertices, e[4].faces, e[4].edges = [v[0], v[4]], [f[3], f[4]], [e[11], e[3], e[0], e[8]] + e[5].vertices, e[5].faces, e[5].edges = [v[5], v[1]], [f[4], f[1]], [e[8], e[0], e[9], e[1]] + e[6].vertices, e[6].faces, e[6].edges = [v[2], v[6]], [f[1], f[2]], [e[1], e[9], e[2], e[10]] + e[7].vertices, e[7].faces, e[7].edges = [v[7], v[3]], [f[3], f[2]], [e[11], e[3], e[10], e[2]] + e[8].vertices, e[8].faces, e[8].edges = [v[4], v[5]], [f[4], f[5]], [e[4], e[5], e[11], e[9]] + e[9].vertices, e[9].faces, e[9].edges = [v[5], v[6]], [f[1], f[5]], [e[5], e[6], e[8], e[10]] e[10].vertices, e[10].faces, e[10].edges = [v[7], v[6]], [f[2], f[5]], [e[7], e[6], e[11], e[9]] e[11].vertices, e[11].faces, e[11].edges = [v[4], v[7]], [f[3], f[5]], [e[4], e[7], e[8], e[10]]