Gaussian Curvature

1# Gaussian Curvature Theory
2def understand_curvature(K):
3    """K = k1 * k2 (the two principal curvatures)"""
4    
5    if K > 0:
6        return "Positive (Synclastic): Like a Sphere. Bends same way in all directions."
7    elif K < 0:
8        return "Negative (Anticlastic): Like a Saddle. Bends up in one direction, down in another."
9    else:
10        return "Zero (Developable): Like a Cylinder or flat plane. Unrolls flat without tearing."

1# Finding the 1-Ring Neighborhood
2def get_vertex_star(mesh, vertex_id):
3    """Returns all faces and vertices immediately connected to a vertex."""
4    
5    # We query the topology graph
6    connected_faces = mesh.topology.get_faces_at_vertex(vertex_id)
7    connected_vertices = mesh.topology.get_neighbors(vertex_id)
8    
9    return connected_faces, connected_vertices

1# Voronoi Area of a Vertex
2def calculate_mixed_voronoi_area(vertex_id, star_faces):
3    """Calculates the surface area 'belonging' to a single vertex."""
4    
5    total_area = 0.0
6    
7    for face in star_faces:
8        # A simple approximation (Barycentric area) is 1/3 of each triangle's area
9        tri_area = calculate_triangle_area(face)
10        
11        # In Meyer's true algorithm, we use the Mixed Voronoi Area
12        # which accounts for obtuse (wide) triangles, but 1/3 is often good enough!
13        total_area += (tri_area / 3.0)
14        
15    return total_area

1# Calculating Angle Deficit
2def calculate_angle_deficit(vertex_id, star_faces):
3    """How much a vertex behaves like a cone vs a saddle."""
4    
5    # Start with 2*PI (a completely flat, 360 degree circle)
6    angle_sum = 2.0 * math.pi
7    
8    for face in star_faces:
9        # Find the angle of this specific triangle at the central vertex
10        theta = calculate_triangle_angle(face, center=vertex_id)
11        
12        # Subtract it from our total
13        angle_sum -= theta
14        
15    # The remainder is the 'Angle Deficit'
16    return angle_sum

1# Calculating Discrete Gaussian Curvature
2def get_gaussian_curvature(vertex_id, mesh):
3    """Combines Angle Deficit with Voronoi Area."""
4    
5    star_faces = mesh.topology.get_faces(vertex_id)
6    
7    # 1. Get the raw angular deficit
8    angle_deficit = calculate_angle_deficit(vertex_id, star_faces)
9    
10    # 2. Get the surface area belonging to this vertex
11    voronoi_area = calculate_mixed_voronoi_area(vertex_id, star_faces)
12    
13    # 3. Normalize!
14    # Without this, a densely tessellated sphere would seem to have 
15    # less curvature per-vertex than a low-poly sphere.
16    K = angle_deficit / voronoi_area
17    
18    return K

1# Vertex Color Mapping
2def apply_curvature_colors(mesh, k_array):
3    """Maps numerical curvature to an RGB gradient."""
4    
5    # K can be anything from -100 to +100 depending on scale.
6    # We need to map this to a 0.0 to 1.0 color gradient.
7    
8    k_min = min(k_array)
9    k_max = max(k_array)
10    
11    for vertex_id, K in enumerate(k_array):
12        # Normalize K to a 0-1 range based on the mesh extremes
13        normalized_k = (K - k_min) / (k_max - k_min)
14        
15        # Look up color in a gradient map (e.g. Blue -> Green -> Red)
16        color = sample_gradient(normalized_k)
17        
18        mesh.vertices[vertex_id].color = color

1# Complete Curvature Pipeline
2def analyze_surface_curvature(mesh):
3    """Full discrete differential geometry analysis."""
4    
5    curvature_data = []
6    
7    for vertex in mesh.vertices:
8        # 1. Topological Analysis (1-Ring)
9        star = get_vertex_star(mesh, vertex.id)
10        
11        # 2. Geometric Analysis
12        deficit = calculate_angle_deficit(vertex.id, star)
13        area = calculate_mixed_voronoi_area(vertex.id, star)
14        
15        # 3. Normalization
16        K = deficit / area
17        curvature_data.append(K)
18        
19    # 4. Rendering
20    apply_curvature_colors(mesh, curvature_data)
21    
22    return mesh