1. Geometric Modeling

2. Volumetric o Collection device obtains regular grid of measurement values Examples: CT, MRI, PET, Ultrasound o Values are interpreted/visualized with color, transparency, texture Example: Marching Cubes Geometric Modeling

3. Procedural o Geometry is calculated by some equation or algorithm and rendered. The algorithms can be as creative as one can imagine, but we can divide them into two types: fractal (natural) and non-fractal (man-made) o Examples:

4. Geometric Modeling Implicit Surface o Geometry is assumed at all points that satisfy equation f(x, y, z) = 0 o Examples: A sphere centered at (a, b, c) of radius r is the set of all points satisfying the equation (x-a) 2 + (y-b) 2 + (z-c) 2 = r 2 o Multiple implicit surfaces can be grouped together to create more complicated shapes.

5. Geometric Modeling Parametric Patch o Geometry is calculated using parametric equations and varying a parametric parameter over some range. o Wavefront uses the term “free-form geometry.” o Examples: Video describing how parametric equations are used with parametric patches(see 1:17-5:20) Video The equation p(u, v) = [ u v k 1 u 2 +k 2 v 2 ] describes the parabola seen here as u and v vary over [0, 1]

6. Geometric Modeling Modeling with Higher-Order Surfaces (Quadrics and Cubics) o Quadrics: Supports common shapes such as circles, ellipses, and parabolas o Examples: o Cubics: General purpose patches allows for favorable continuity conditions o Examples:

7. Other Modeling Methods o Constructive Solid Geometry (CSG) builds intricate shapes from simple ones using set operations. This type of modeling is common to CAD (such as with the SolidWorks modeler), because interiors may be defined easily Example: o Terrain Generators are a procedural modeling tool, but often customized for generating “terrain-like” models (i.e., usually controlled by some type of height map) Geometric Modeling

8. Modeling with Planer Polygons o This is by far the most common case in (non-engineering) computer graphics today Engineering models prefer CSG or bi-cubic patches for analysis and manufacture Commodity display hardware is optimized for planer polygons Further optimized for triangles (and sometime quads). o Types of Polygonal Models Extrusions Replications Symmetric o Bi-symmetry o Surface of Revolution Compositions of Simpler Surfaces

9. 3D Modeling Terminology Coordinate System: o XYZ (RGB) o Right Handed o Handedness effects rotation Pos rotation follows finger curl Reference Frames o Model o World o Eye o View Model World Eye View

10. 3D Modeling Terminology Vertex Edge Normal o Computation Rule Computation Rule Face o Front Face o Back Face o CCW Traversal Rule CCW Traversal Rule Mesh o Manifold Subdivision o Triangulation Simplification/Decimation Transformations o Translate o Rotate o Scale o Shear o Mirror

11. Rules for Oriented Faces CCW Rule: An in-order traversal of the vertices of a face will be counter-clockwise for a front-facing polygon. Normal Computation Rule: The normal of a face can be calculated by the cross product of any three, convex vertices of the face: o E.g., (v 5 – v 3 ) X (v 1 – v 3 ) o The normal is typically normalized (made unit length) by dividing by its length, denoted || (v 5 – v 3 ) X (v 1 – v 3 )|| v1v1 v2v2 v4v4 v3v3 v5v5 v6v6 v7v7

12. Wavefront obj File Example