CSCE 441 Lecture 28

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Implicit Representations of Shape

No polygons; just functions defining solution points

Advantages

• No topology to maintain
• Always defines a closed surface
• Inside/Outside test is easy
• CSG operations
  • Union: take min of each function
  • Intersection: take max of each function
  • Subtraction: $\left\{f(x)\right\}-\left\{g(x)\right\}=\max {f(x),-g(x)}$

Disadvantages

• hard to render - no polygons
• creating polygons
• Arbitrary shapes hard to represent

2D Polygon Generation

Sample over grid and get large 2D array of positive/negative values.

Place vertices on edges where function changes sign: use linear interpolant of values at each edge endpoint (higher magnitude means further away from zero set)

3D Polygon Generation

Use 3D grid made from cubes instead of 2D grid

8 vertices, each vertex has 2 possibilities (positive or negative), so there are $2^{8}$ possible configurations.

There are 15 equivalence classes: