Isotropic Remeshing

2 min read•1 page

Isotropic Remeshing (Botsch & Kobbelt) iteratively transforms irregular, stretched triangles into nearly equilateral triangles of uniform target edge length $L$.

Split & Collapse Criteria:

1. Split Long Edges: Any edge with length $> \frac43 L$ is bisected at its midpoint. 2. Collapse Short Edges: Any edge with length $< \frac45 L$ is collapsed to an endpoint. 3. Target Uniformity: Prevents needle triangles and extreme area variations.

python

1def isotropic_remesh_pass(mesh, target_length):
2    high_threshold = (4.0 / 3.0) * target_length
3    low_threshold = (4.0 / 5.0) * target_length
4
5    # 1. Split edges longer than high_threshold
6    for edge in mesh.edges():
7        if edge.length() > high_threshold:
8            mesh.split_edge(edge)
9
10    # 2. Collapse edges shorter than low_threshold
11    for edge in mesh.edges():
12        if edge.length() < low_threshold:
13            mesh.collapse_edge(edge)

Target Length (L)

2.50

3 min read•1 page

Valence Optimization flips non-delaunay or irregular diagonal edges so that every interior vertex approaches the ideal valence of 6 (the optimal hexagonal packing topology).

Valence Flip Criterion:

1. Target Valence 6: Regular triangulations have 6 equilateral neighbors per vertex. 2. Cost Metric: Minimizes sum of squared valence errors (valence - 6)^2 before and after an edge swap. 3. Geometric Invariant: Flips are rejected if the resulting flipped triangles invert or fold.

python

1def flip_edges_for_valence(mesh):
2    """Flip edges if doing so brings vertex valences closer to target 6."""
3    target_valence = 6
4    
5    for edge in mesh.interior_edges():
6        v1, v2 = edge.vertices
7        v3, v4 = edge.opposite_vertices()
8        
9        # Calculate current squared deviation from valence 6
10        current_cost = (
11            (v1.valence - target_valence)**2 +
12            (v2.valence - target_valence)**2 +
13            (v3.valence - target_valence)**2 +
14            (v4.valence - target_valence)**2
15        )
16        
17        # After flipping: v1 and v2 lose 1 edge; v3 and v4 gain 1 edge
18        flipped_cost = (
19            (v1.valence - 1 - target_valence)**2 +
20            (v2.valence - 1 - target_valence)**2 +
21            (v3.valence + 1 - target_valence)**2 +
22            (v4.valence + 1 - target_valence)**2
23        )
24        
25        if flipped_cost < current_cost:
26            mesh.flip_edge(edge)

Flip Diagonal Edge