surfaces/geometry.py

# import scipy

'''
http://en.wikipedia.org/wiki/Polygon_mesh

Polygon meshes may be represented in a variety of ways, using different methods to store the vertex, edge and face data. These include:
    Face-vertex meshes: A simple list of vertices, and a set of polygons that vertex to the vertices it uses.
    Winged-edge meshes, in which each edge vertices to two vertices, two faces, and the four (clockwise and counterclockwise) edges that touch it. Winged-edge meshes allow constant time traversal of the surface, but with higher storage requirements.
    Half-edge meshes: Similar to winged-edge meshes except that only half the edge traversal information is used.
    Quad-edge meshes, which store edges, half-edges, and vertices without any reference to polygons. The polygons are implicit in the representation, and may be found by traversing the structure. Memory requirements are similar to half-edge meshes.
    Corner-tables, which store vertices in a predefined table, such that traversing the table implicitly defines polygons. This is in essence the "triangle fan" used in hardware graphics rendering. The representation is more compact, and more efficient to retrieve polygons, but operations to change polygons are slow. Furthermore, corner-tables do not represent meshes completely. Multiple corner-tables (triangle fans) are needed to represent most meshes.
    Vertex-vertex meshes: A "VV" mesh represents only vertices, which vertex to other vertices. Both the edge and face information is implicit in the representation. However, the simplicity of the representation allows for many efficient operations to be performed on meshes.

        
    Face-vertex meshes represent an object as a set of faces and a set of vertices. This is the most widely used mesh representation, being the input typically accepted by modern graphics hardware.
                v4            v5
                 *-----e8-----*
                 |            |
                 |            |
                e|4    f4    e|5
                 |            |
   v4           v|0          v|1           v5
    *-----e4-----*-----e0-----*-----e5-----*
    |            |            |            |
    |            |            |            |
 e11|     f3    e|3    f0    e|1    f1    e|9
    |            |            |            |
    |            |            |            |
    *-----e7-----*-----e2-----*-----e6-----*
   v7           v|3          v|2           v6
                 |            |
                e|7    f2    e|6
                 |            |
                v|7          v|6
                 *-----e10----*
                 |            |
                 |            |
                e|11   f5    e|9
                 |            |
                 |            |
                 *-----e8-----*
                v4            v5
    v0 - <0,1,0>
    v1 - <1,1,0>
    v2 - <1,0,0>
    v3 - <0,0,0>
    v4 - <0,1,1>
    v5 - <1,1,1>
    v6 - <1,0,1>
    v7 - <0,0,1>
    
    face list
    f0 - e0, e1, e2, e3
    f1 - e1, e5, e9, e6
    f2 - e2, e6, e10, e7
    f3 - e4, e3, e7, e11
    f4 - e8, e5, e0, e4
    f5 - e10, e9, e8, e11
    
    winged edges ordered by face, then by vertex reference
    edge list
    e0 - v0, v1; f0, f4; e3, e1, e4, e5
    e1 - v1, v2; f0, f1; e0, e2, e5, e6
    e2 - v3, v2; f0, f2; e3, e7, e1, e6
    e3 - v0, v3; f3, f0; e4, e7, e0, e2
    e4 - v4, v0; f3, f4; e11, e3, e0 e8
    e5 - v5, v1; f4, f1; e8, e0, e9, e1
    e6 - v2, v6; f1, f2; e1, e9, e2, e10
    e7 - v7, v3; f3, f2; e11, e3, e10, e2
    e8 - v4, v5, f4, f5; e4, e5, e11, e9
    e9 - v5, v6; f1, f5; e5, e6, e8, e10
    e10 - v7, v6; f2, f5; e7, e6, e11, e9
    e11 - v4, v7; f3, f5; e4, e7, e8, 10
    
    vertex list
    v0 - e0, e3, e4
    v1 - e0, e5, e1
    v2 - e1, e6, e2
    v3 - e2, e7, e3
    v4 - e4, e11, e8
    v5 - e5, e9, e8
    v6 - e2, e9, e10
    v7 - e7, e10, e11
    
'''

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):
        self.x = x
        self.y = y
        self.z = z
        self.edges = []
    
    def __repr__(self):
        return "<%.2f, %.2f, %.2f>" % (self.x, self.y, self.z)

class Edge(object):
    '''
    '''
    def __init__(self):
        self.vertices = []
        self.faces = []
        self.edges = []
        self.__edgeVertex = None
        self.__subEdges = []
    
    def neighborFace(self, neighborFace):
        if neighborFace == self.faces[0]:
            return self.faces[1]
        else:
            return self.faces[0]
    
    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.
        '''
        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):
        
        return 
    
class Face(object):
    '''
    A face is a closed set of edges, in which a triangle face has three edges, and a quad face has four edges.
    '''
    def __init__(self):
        self.edges = []
        self.__centroid = None
        self.__interiorEdges = []
        self.__subFaces = []
        
    def __getCentroid(self):
        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 __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):
    '''
    Face splitting should happend on the polygon level(?).  It doesn't make sense to split just one face since
        it needs to average vertices with all adjoinging faces
    '''
    
    def __init__(self):
        self.vertices = []
        self.edges = []
        self.faces = []
    
    def catmullClarkRefine(self):
        '''
        For each face, add a face vertex
            Set each face vertex to be the centroid of all original vertices for the respective face.
        For each edge, add an edge vertex.
            Set each edge vertex to be the average of the two neighbouring face vertices and its two original endvertices.
        For each face vertex, add an edge for every edge of the face, connecting the face vertex to each edge vertex for the face.
        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)
                for edge in subFace.edges:
                    edges.append(edge)
                    for vertex in edge.vertices:
                        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.........        
        # (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)
        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))
    v.append(Vertex(0.0, 0.0, 1.0))

    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())

    f = []
    f.append(Face())
    f.append(Face())
    f.append(Face())
    f.append(Face())
    f.append(Face())
    f.append(Face())

    # 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]]

    # 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]]

    #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]]

    # just to prove to myself that the objects are the same, this is what years of pass by value have done to me...

    # 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 id(v[0].x)
    # print id(e[0].vertices[0].x)
    # print id(f[0].edges[0].vertices[0].x)

    # 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

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()