GAMES101 Lecture 10 - Geometry 1 (Introduction)

I. Application of Textures

Refer to Lecture09.md.

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Implicit: Points satisfy some specified relationship. Example:
$f (x, y, z) = 0$
- Pros:
  - Compact description
  - Certain queries easy (inside object, distance to surface)
  - Good for ray-to-surface intersection
  - For simple shapes, exact description / no sampling error
  - Easy to handle changes in topology
- Cons:
  - Difficult to model complex shapes
- Examples:
  - Algebraic Surfaces
    $x^{2} + y^{2} + z^{2} = 1$
  - Constructive Solid Geometry: Combine implicit geometry via Boolean operations
  - Level Set Methods: Store a grid of values approximating a function. Surface is found where interpolated values equal zero
  - Distance Functions: Given minimum distance (could be signed distance) from anywhere to object
    - Blend a moving boundary
    - Blend any two distance functions
  - Fractals: Exhibit self-similarity, detail at all scales

Explicit: All points are given directly or via parameter mapping. Example:
$f : R^{2} \to R^{3}; (u, v) \to (x, y, z)$
- Easy to sample
- Hard to test inside/outside
- Examples:*
  - Point Cloud
    - Easiest representation
    - Useful for large datasets
    - Often converted into polygon mesh
    - Difficult to draw in under-sampled regions
  - Polygon Meshes
    - Store vertices & polygons (often triangles or quads)
    - Easier to do processing / simulation, adaptive sampling
    - More complicated data structures
    - Perhaps the most common representation in graphics
    - Format Example:
      - The Wavefront Object File (.obj) Format
        Text file specifying vertices, normals ,texture coordinates and their connectivity (by specifying for example the triangular face, with normals and texture coordinates on three vertices)
  - Bezier Surfaces
  - Polygon Mesh
  - Subdivision Surfaces
  - NURBS
- ...