Harmonic & Conformal UV

3 min read•1 page

Tutte's Invariant:

1. Convex Boundary Condition: Outer mesh boundary pinned along a convex polygon (e.g. circle). 2. Harmonic Mean Value: Every internal UV coordinate equals the average of its 1-ring neighbors. 3. Zero Triangle Inversion: Determinant of every triangle's Jacobian remains strictly positive.

python

1import numpy as np
2
3def tutte_barycentric_mapping(boundary_indices, boundary_uvs, adjacency_list):
4    """
5    Tutte's Spring Embedding Theorem (1963).
6    Maps 3D mesh boundary to a convex 2D polygon (e.g. circle or square).
7    Interior vertices satisfy harmonic equilibrium:
8    u_i = (1 / d_i) * sum_{j in N(i)} u_j.
9    Guarantees a strictly bijective, flip-free, 1-to-1 planar embedding.
10    """
11    n = len(adjacency_list)
12    A = np.zeros((n, n))
13    bx = np.zeros(n)
14    by = np.zeros(n)
15    
16    for i in range(n):
17        if i in boundary_indices:
18            A[i, i] = 1.0
19            idx = boundary_indices.index(i)
20            bx[i] = boundary_uvs[idx][0]
21            by[i] = boundary_uvs[idx][1]
22        else:
23            deg = len(adjacency_list[i])
24            A[i, i] = 1.0
25            for j in adjacency_list[i]:
26                A[i, j] = -1.0 / deg
27                
28    u = np.linalg.solve(A, bx)
29    v = np.linalg.solve(A, by)
30    return list(zip(u, v))

Tutte's Barycentric Mapping (Harmonic Spring Embedding): Unwrapping a double-curved 3D mesh onto flat 2D UV texture space easily causes triangles to overlap or invert. In 1963, W. T. Tutte proved a foundational mathematical theorem: if the boundary of a 3-connected planar graph is mapped to the vertices of a strictly convex 2D polygon, and every internal vertex is placed at the weighted centroid of its neighbors (spring equilibrium), the resulting 2D parameterization is guaranteed to be 100% bijective and flip-free.

0=3D Hemispherical Mesh, 1=2D Tutte UV

1.00

5 min read•2 pages

LSCM Conformal Invariant:

1. Cauchy-Riemann Energy: Minimizes gradient mismatch ∇ u - J ∇ v = 0 where J is 90° rotation. 2. Free Boundary Relaxation: Boundary is free to assume its natural shape with only two pinned vertices. 3. Zero Angular Shearing: Texture grid squares remain strictly orthogonal and undistorted.

python

1import numpy as np
2
3def solve_lscm_uv(triangles, vertices, uv_coords):
4    """
5    Evaluates Least Squares Conformal Maps (LSCM, Levy et al. 2002) energy across a 3D mesh:
6    E_LSCM = sum_T 0.5 * Area(T) * || grad(u) - J * grad(v) ||^2
7    where J = [[0, -1], [1, 0]] rotates the 2D gradient by 90 degrees (Cauchy-Riemann check).
8    """
9    J = np.array([[0.0, -1.0], [1.0, 0.0]])
10    total_conformal_energy = 0.0
11
12    for i0, i1, i2 in triangles:
13        p0, p1, p2 = [np.asarray(vertices[k], dtype=float) for k in (i0, i1, i2)]
14        uv0, uv1, uv2 = [np.asarray(uv_coords[k], dtype=float) for k in (i0, i1, i2)]
15
16        # Local orthonormal 2D basis (e1, e2) in the 3D triangle plane
17        e1 = p1 - p0
18        x1 = np.linalg.norm(e1)
19        e1 /= x1
20        n = np.cross(e1, p2 - p0)
21        area = 0.5 * np.linalg.norm(n)
22        e2 = np.cross(n / np.linalg.norm(n), e1)
23        x2, y2 = np.dot(p2 - p0, e1), np.dot(p2 - p0, e2)
24
25        # Barycentric gradient vectors in local (x, y) triangle frame
26        grad_B = np.array([[y2 - 0.0, 0.0 - y2, 0.0],
27                           [0.0 - x2, x2 - x1, x1]]) / (2.0 * area)
28        grad_u = grad_B @ np.array([uv0[0], uv1[0], uv2[0]])
29        grad_v = grad_B @ np.array([uv0[1], uv1[1], uv2[1]])
30        total_conformal_energy += 0.5 * area * float(np.sum((grad_u - J @ grad_v) ** 2))
31    return total_conformal_energy

Least Squares Conformal Maps (LSCM): While Tutte mapping forces boundaries into an arbitrary rigid polygon (distorting texture angles), LSCM (Lévy et al. 2002) minimizes the deviation from the Cauchy-Riemann equations. Conformal maps preserve local angles: microscopic circles on the 3D surface map to perfect 2D circles in texture space without elliptical shearing, making it the industry standard for UV texture unwrapping.

Pinned Vertex Distance

2.40

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