Polygons
 | Most basic representation for 3D systems (previously triangles) |
| Made up of vertices and edges |
| Order in which vertices are traversed determines direction of normal |
Normals
| Used to determine surface lighting (amount based on dot product with light
direction) |
| If polygon is planar, can calculate normal as cross product of edge
vectors |
| If not, approximate by "averaging" using Newell's method |
| Can be calculated per polygon, flat shading, or per vertex, interpolated
shading |
Mathematical Representations of Surfaces
| Implicit or functional form: F(x, y, z) = 0 |
| Explicit or parametric form: P(u, v) = (X(u, v), Y(u, v), Z(u, v)) |
| Example: sphere |
Techniques for Generating Surfaces
| Typically start with 2D polygon or curve and extend in 3D |
| Polyhedra, any closed collection of planar polygons
Since many vertices are shared, avoid redundancy by separating vertex, normal, and face data
|
| Extrusions, any 2D shape that is pulled or swept through 3D space
 | Segmented tubes
 |
| Ruled surfaces
 |
| Surfaces of revolution
 |
|
| Quadrics, 3D analogs of conic sections
 |
| Superquadrics, quadrics extended with "bulge" factors to produce
greater variety
 |
| Constructive Solid Geometry (CSG), combine above surfaces via union,
intersection, difference
 |
Tessellating Surfaces
| Goal: generate fewest number of polygons |
| Variety of ways to polygonalize ideal surface |
| Simplest: given function of surface, sample at given intervals
 |
|
Marching cubes, divide 3D space into cubes, sample at specific points, connect
samples
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References
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