Planar Quad (PQ) Meshes

1import numpy as np
2
3def evaluate_quad_planarity(v0, v1, v2, v3):
4    """
5    Computes planarity error of quadrilateral (v0, v1, v2, v3).
6    A quad is strictly planar if the scalar triple product is zero:
7    vol = det([v1 - v0, v2 - v0, v3 - v0]) = 0.
8    Normalized planarity metric: distance from v3 to plane(v0, v1, v2).
9    """
10    e1 = v1 - v0
11    e2 = v2 - v0
12    normal = np.cross(e1, e2)
13    norm_len = np.linalg.norm(normal)
14    if norm_len < 1e-9:
15        return 0.0  # Degenerate
16    unit_norm = normal / norm_len
17    
18    # Distance of 4th vertex from plane
19    dist_v3 = abs(np.dot(v3 - v0, unit_norm))
20    diag_avg = (np.linalg.norm(v2 - v0) + np.linalg.norm(v3 - v1)) * 0.5
21    relative_planarity = dist_v3 / diag_avg
22    return relative_planarity

1import numpy as np
2
3def is_conical_vertex(vertex, neighbor_ring, face_normals, tol=1e-4):
4    """
5    Evaluates whether a degree-4 quad mesh vertex is conical (Pottmann & Wallner 2006):
6    1. Opposite interior corner angles sum equally: omega_1 + omega_3 == omega_2 + omega_4.
7    2. Computes the vertex cone axis A such that A . n_i = cos(theta) is constant,
8       allowing exact constant-distance face offsets v_offset = v + (d / cos(theta)) * A.
9    """
10    rays = [ (p - vertex) / np.linalg.norm(p - vertex) for p in neighbor_ring ]
11    angles = [ np.arccos(np.clip(np.dot(rays[i], rays[(i + 1) % 4]), -1.0, 1.0)) for i in range(4) ]
12    angle_balance = abs((angles[0] + angles[2]) - (angles[1] + angles[3]))
13
14    # Solve for cone axis A satisfying (n_i - n_0) . A = 0
15    N = np.asarray(face_normals[:3], dtype=float)
16    axis = np.cross(N[1] - N[0], N[2] - N[0])
17    axis /= np.linalg.norm(axis) + 1e-12
18    if np.dot(axis, N[0]) < 0:
19        axis = -axis
20    cos_theta = np.dot(axis, N[0])
21    return (angle_balance < tol), axis, float(cos_theta)