Spatial orientation and polygon tables

Spatial orientation of a surface

The orientation of a polygon surface in space is described by its plane equation Ax + By + Cz + D = 0, where the normal vector N = (A, B, C) is perpendicular to the surface. For three vertices taken in counter-clockwise order, N = (V2 − V1) × (V3 − V1) and D = −N · V1.

Significance

1. Inside/outside test: a point (x, y, z) is in front of the surface if Ax + By + Cz + D > 0 and behind it if < 0.
2. Visible-surface detection: back faces are found from the sign of V·N (V = viewing vector).
3. Shading: the angle between N and the light direction decides the diffuse intensity (cosθ = N·L).
4. Clipping and collision tests: which side of a plane a point lies on.
5. Rendering: normals are interpolated for Gouraud and Phong shading.

Polygon tables

Polygon tables (vertex, edge and surface tables) store the geometry in a form that is easy to process. The plane coefficients A, B, C, D of each surface are computed once from the vertex table and stored in the surface table, so they are available for visibility and shading.

Example: triangle P1(1,0,0), P2(0,1,0), P3(0,0,1)

Vertex: V1 (1,0,0)  V2 (0,1,0)  V3 (0,0,1)
Edge:   E1 V1V2  E2 V2V3  E3 V3V1
Surface S1: E1 E2 E3,  N = (1,1,1), D = -1

N = (V2 − V1) × (V3 − V1) = (−1,1,0) × (−1,0,1) = (1,1,1). For the eye at (5,5,5): 5 + 5 + 5 − 1 = 14 > 0, so the front face is visible.