Representing 3D Objects

Common 3D representations:

1. Polygon Meshes - dominant for real-time (Unit III).
2. Sweep Representations - rotate or translate a 2D shape (lathe / extrusion).
3. Constructive Solid Geometry (CSG) - boolean operations (union, intersection, difference) on primitives (cube, sphere, cylinder).
4. Voxels - 3D grid of cells; used in medical imaging, Minecraft.
5. Point Clouds - sets of 3D samples from LIDAR or photogrammetry.
6. Implicit Surfaces - f(x,y,z) = 0; smooth blobs (metaballs), signed distance fields.
7. Parametric Surfaces - Bezier patches, NURBS, subdivision surfaces (Catmull-Clark).

What Is a Projection?

A projection maps 3D points to a 2D plane (the view plane / projection plane) along projectors (rays). Two big families:

1. Parallel: projectors are parallel; preserves parallel lines, distorts realism but preserves measurements.
2. Perspective: projectors converge to a center of projection (eye); produces foreshortening; matches human vision.

Parallel vs Perspective Comparison

View Plane Variables

• Center of projection (COP): at infinity for parallel, finite for perspective.
• Direction of projection (DOP): vector along projectors (parallel only).
• View plane normal (VPN): orientation of the projection plane.
• View reference point (VRP): an anchor on the plane.
• View up vector (VUV): defines roll.

Projection Matrix Sketch

Parallel orthographic on z=0 plane: drop the z coordinate. M_par = [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 0 ] [ 0 0 0 1 ]

Perspective on z=d plane with COP at origin: M_per = [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 1/d 0 ] After multiplying by [x,y,z,1]^T we get [x, y, z, z/d]; dividing by w gives (xd/z, yd/z, d) on the view plane. The 1/d in the bottom row is the "perspective divide" that makes far things smaller.

What's Coming

Next lessons drill into each subtype, derive their matrices, and compare visual results.