GAMES101 Lecture 12 - Geometry 3 (Mesh Operations and Shadow Mapping)

GAMES101_Lecture_12.pdf

Turing Award Winners (in CG)

Make Computer Graphics great again

• Ed Catmull

• Pat Hanradan

I. Mesh Operations: Geometry Processing

• Mesh Subdivision (Upsampling): Increase resolution.

  • Loop Subdivision: New/old vertices updated differently

    • Procedure:

      • 1. Insert new vertices: Split each triangle into four.

        

      • 2. Update positions of vertices

        • For new vertices:

          Update the position to:

          $$\frac{3}{8}(A+B)+\frac{1}{8}(C+D)$$

          

        • For old vertices:

          Let $n$$n$ be the degree of the vertex, and $p_i,i>0$$p_i, i>0$ be the old vertices surround the current vertex $p_0$$p_0$. Then the new position is:

          $$(1-nu)p_0+\sum _{i>0}u{p}_i,\text{where }u=3/16\text{ if }n=3\text{, and }3/(8n)\text{ otherwise}$$

          

     

  • Catmull-Clark Subdivision (General Mesh)

    • Extraordinary vertex: vertex that has $\text{degree}!=4$$\text{degree} != 4$

    • Non-quad face

    • Procedure:

      • 1. Add new vertices

        • Add vertex at the center of each face

        • Add midpoint on each edge

        • Connect all new vertices

      • 2. Update the Mesh

        • face point, edge point, vertex point...

          

           

• Mesh Simplification (Downsampling): Decrease resolution; Try to preserve shape/appearance.

  • Edge Collapsing: Contract two vertices into one.

    

  • Quadric Error Metrics: New vertex should minimize its sum of square distance (L2 distance) to previously related triangle planes.

  • (Garland & Heckbert 1997):

    • Approximate distance to surface as sum of distances to planes containing triangles

    • Iteratively collapse edge with smallest score

• Mesh Regularization: Improve the quality of the mesh; Try to make the shape of each triangle regularized.

II. Shadow Mapping

Refer to Lecture09.md.