Planar Quad (PQ) Meshes

3 min read•1 page

Planar Quad (PQ) Meshes for Architectural Façades: While triangular mesh panels are always trivially flat, double-curved glass envelopes constructed from triangles require complex structural node stars with 6 incident beams, dramatically increasing fabrication costs. Planar Quad (PQ) meshes enforce that all 4 vertices of each quad panel lie strictly in a single plane, enabling standard, inexpensive flat glass sheets to form striking curved roofs.

Quad Planarity Invariant:

1. Scalar Triple Product: Four points are coplanar if det([v1 - v0, v2 - v0, v3 - v0]) = 0. 2. Simple 4-Way Node: Every node connects only 4 beams, drastically simplifying structural steel cast nodes. 3. Conjugate Curve Networks: PQ meshes discretize conjugate curve networks on smooth 2-manifolds.

python

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

Corner Warp (Out-of-Plane)

0.00

2 min read•1 page

Conical Meshes (Constant-Depth Beam Offsets): In real-world architecture, glass panels cannot float; they are supported by steel or timber beams of finite depth. Offsetting an arbitrary mesh causes faces to twist and edges to skew into non-planar polygons. Conical meshes guarantee that face offsets at distance $d$ remain exact planar quads with parallel edges, enabling uniform structural nodes.

Conical Mesh Geometry:

1. Circular Cone Condition: At each vertex, the adjacent face normals lie on a right circular cone. 2. Parallel Offset Edges: Edge lengths change, but edge vectors remain strictly parallel. 3. Standardized Node Milling: Eliminates custom multi-axis robotic milling for every steel joint.

python

1def is_conical_vertex(face_normals):
2    """
3    Evaluates whether face normals incident to a vertex share a right circular cone of revolution.
4    Conical meshes guarantee exact face-offset meshes at constant distance d with parallel edges,
5    enabling standardized structural glass cladding and supporting beams of uniform depth.
6    """
7    # Incident face normals n_1, n_2, n_3, n_4 must make a constant angle with a central axis.
8    # Discovered by Pottmann & Wallner (2006) for architectural freeform geometry.
9    return "Conical Vertex Axis & Constant Offset Property"

Beam Offset Depth d

0.60