Mesh Topology & Decimation

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Connectivity Links:

1. Twin Reference: Links opposite half-edges, allowing traversal between adjacent faces. 2. Next Reference: Links to the next half-edge in the clockwise/counter-clockwise face loop. 3. Origin Reference: Identifies the starting vertex of the half-edge.

python

1class HalfEdge:
2    def __init__(self):
3        self.origin = None    # Vertex at start of half-edge
4        self.twin = None      # Opposite half-edge sharing same edge
5        self.next = None      # Next half-edge in face boundary loop
6        self.face = None      # Face this half-edge bounds
7
8class Vertex:
9    def __init__(self, pos):
10        self.position = pos   # (x, y, z) coordinate
11        self.half_edge = None # Outgoing half-edge reference

To edit mesh topology efficiently, we need a structure that allows local traversals. The Half-Edge structure splits every edge into two directed halves pointing in opposite directions, linking vertices, faces, and edges together.

Highlight (0=Base Mesh, 1=Twin Half-Edges, 2=Next/Prev Loop)

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2 min read•1 page

Local Operations:

1. Edge Collapse: Merges two vertices, deleting one edge and its two adjacent faces. 2. Edge Split: Inserts a vertex in the middle of an edge, splitting adjacent triangles into four. 3. Edge Flip: Re-routes a shared edge between two triangles to the opposite vertices.

python

1def collapse_edge(mesh, half_edge):
2    v0 = half_edge.origin
3    v1 = half_edge.twin.origin
4
5    # 1. Update outgoing/incoming pointers around v1 to v0
6    for edge in incoming_edges(v1):
7        edge.target = v0
8
9    # 2. Remove the collapsed edge and its twin
10    # 3. Remove the two adjacent faces that become degenerate
11    remove_vertex(mesh, v1)
12    remove_edge(mesh, half_edge)
13    remove_face(mesh, half_edge.face)
14    remove_face(mesh, half_edge.twin.face)

Mesh simplification algorithms reduce polygon counts by applying local topological operations. The most fundamental of these is the Edge Collapse, which merges two vertices into one.

Mesh Operation

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