Modeling
- How to represent objects for 3D graphics?
- How to construct such representations quickly and/or automatically?
- How to manipulate such representations?
Variety of Representations
- Raw Data
- point cloud - unstructured set of 3D point samples
- range image - depth data (acquired from range scanner)
- unconnected polygons - created interactively?
- Surfaces
- mesh - connected set of polygons
- subdivision - coarse mesh + subdivision rule
- parametric - geometric equation
- implicit - points satisfying function: F(x, y, z) = 0
- Solids
- voxels - uniform grid of sample data (acquired from MRI, CAT, etc)
- BSP trees - tree representation of 3D data
- CSG - hierarchy of boolean operations applied to simple shapes
- High-level structures
- scene graph - union of objects at leaf nodes
- skeleton - graph of curves with radii
- application specific
Goals
- Accuracy
- Conciseness
- Intuitive specification
- Affine invariant
- Natural parameterization
- Efficient display
- Efficient intersections
- Arbitrary topology
- Guaranteed continuity
- Local support
Polygon Mesh
- Most basic representation for 3D systems (usually triangles)
- Made up of vertices, edges, and faces
- 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
- Platonic solids
- Prisms
- 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
References
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