Marching Cubes

1# Scalar Fields to Meshes
2def extract_isosurface(voxel_grid, threshold):
3    """The magic behind Medical MRI scans and Minecraft mods."""
4    
5    mesh = Mesh()
6    
7    # 1. Loop through every single box in the grid
8    for voxel in voxel_grid:
9        # 2. Does the surface pass through this specific box?
10        if voxel_contains_surface(voxel, threshold):
11            
12            # 3. If yes, generate triangles for just this box!
13            triangles = marching_cubes(voxel)
14            mesh.add(triangles)
15            
16    return mesh

1# Classifying the 8 Corners
2def get_cube_index(cube_corners, threshold):
3    """Generates an 8-bit binary number (0 to 255)."""
4    
5    cube_index = 0
6    
7    # Check all 8 corners of the cube
8    for i in range(8):
9        corner_value = cube_corners[i].value
10        
11        # If the corner is INSIDE the surface
12        if corner_value < threshold:
13            # Flip the i-th bit to 1!
14            # Example: 1 << 0 = 1, 1 << 1 = 2, 1 << 2 = 4...
15            cube_index |= (1 << i)
16            
17    return cube_index

1# The Lookup Table (LUT)
2def lookup_triangles(cube_index):
3    """Avoids calculating geometry in real-time."""
4    
5    # We pre-calculate all 256 possible triangle configurations
6    # and store them in a massive Array/Dictionary when the game loads.
7    
8    # TRIANGLE_TABLE[0] = [] # Completely outside, no triangles
9    # TRIANGLE_TABLE[255] = [] # Completely inside, no triangles
10    # TRIANGLE_TABLE[1] = [ [0, 8, 3] ] # One corner cut off
11    
12    # We simply use our 8-bit index to look up the answer instantly!
13    intersected_edges = TRIANGLE_TABLE[cube_index]
14    
15    return intersected_edges

1# Finding the Exact Intersection
2def interpolate_edge(corner_A, corner_B, threshold):
3    """Slides the vertex along the edge for a perfect fit."""
4    
5    # 1. Get the scalar values at both ends of the edge
6    val_A = corner_A.value
7    val_B = corner_B.value
8    
9    # 2. How far along the edge is our threshold?
10    # e.g., if A=0.2, B=0.8, and threshold=0.5, then ratio = 0.5 (exactly halfway)
11    ratio = (threshold - val_A) / (val_B - val_A)
12    
13    # 3. Use that ratio to mix the XYZ coordinates
14    x = corner_A.x + ratio * (corner_B.x - corner_A.x)
15    y = corner_A.y + ratio * (corner_B.y - corner_A.y)
16    z = corner_A.z + ratio * (corner_B.z - corner_A.z)
17    
18    return Point(x, y, z)

1# Connecting the Dots
2def generate_cube_geometry(intersected_edges, corner_values):
3    """Turns the edge dots into solid triangles."""
4    
5    triangles = []
6    
7    # The Lookup Table returns edges in sets of 3.
8    # e.g., [Edge_0, Edge_8, Edge_3, Edge_1, Edge_9, Edge_2]
9    
10    # We step through the list 3 edges at a time
11    for i in range(0, len(intersected_edges), 3):
12        
13        # Interpolate exact XYZ for the 3 edges
14        pt1 = interpolate_edge(intersected_edges[i])
15        pt2 = interpolate_edge(intersected_edges[i+1])
16        pt3 = interpolate_edge(intersected_edges[i+2])
17        
18        # Connect them to form a triangle!
19        triangles.append(Triangle(pt1, pt2, pt3))
20        
21    return triangles

1# Calculating Smooth Normals
2def calculate_normal(x, y, z):
3    """How to make a jagged mesh look perfectly smooth."""
4    
5    # 1. We look at the grid values immediately around our point
6    # We check +X and -X, +Y and -Y, +Z and -Z
7    
8    # 2. We calculate the difference (Gradient)
9    nx = value_at(x - 1, y, z) - value_at(x + 1, y, z)
10    ny = value_at(x, y - 1, z) - value_at(x, y + 1, z)
11    nz = value_at(x, y, z - 1) - value_at(x, y, z + 1)
12    
13    # 3. Combine them into a Vector and Normalize it (length = 1.0)
14    normal = Vector3(nx, ny, nz).normalize()
15    
16    return normal

1# Full Marching Cubes Pipeline
2def generate_bunny_mesh(voxel_grid, threshold):
3    """From blocky Voxels back to a smooth Mesh."""
4    
5    mesh = Mesh()
6    
7    # Run the algorithm on the grid
8    for voxel in voxel_grid:
9        # Get the 8 corner values
10        corners = get_corners(voxel)
11        
12        # Get the 8-bit index
13        idx = get_cube_index(corners, threshold)
14        
15        # Look up which edges are cut
16        edges = lookup_triangles(idx)
17        
18        # Interpolate exact points and form triangles
19        triangles = generate_cube_geometry(edges, corners)
20        
21        mesh.add(triangles)
22        
23    # Calculate smooth lighting normals
24    mesh.calculate_smooth_normals()
25    
26    return mesh