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Procedural Models

Procedural Models. How to easily generate complex data? Data from parameterized procedure and/or digitized Data amplification Ray trace directly v. convert to polygonal models and use z-buffer. Basic Procedural Approaches. composition from primitive shapes extrusion

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Procedural Models

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  1. Procedural Models • How to easily generate complex data? • Data from parameterized procedure and/or digitized • Data amplification • Ray trace directly v. convert to polygonal models and use z-buffer

  2. Basic Procedural Approaches • composition from primitive shapes • extrusion • surface of revolution • lofting • sweep operator

  3. Composition Procedurally generated Controlled randomness

  4. Extrusion

  5. Surface of revolution

  6. Lofting

  7. Sweep Operator Combines extrusion, lofting, and surface of revolution

  8. More sophisticated approaches • CSG • Subdivision surfaces • Curves & surfaces • Fractals • Implicit functions

  9. CSG:constructivesolid geometry • primitive shapes • union, difference, intersection • tree structure representation • Ray trace directly • Evaluate to polyhedron with Boolean operators • OpenGL rendering using stencil buffer • http://www.opengl.org/resources/code/samples/advanced/ advanced97/notes/node11.html

  10. Subdivision surfaces What is resulting limit surface? • What is continuity of limit surface? • Initial coursely defined geometry • Refine geometry by • rounding corners, • subdividing faces and edges, • smoothing faces

  11. Subdivision surfaces • Round corners • Catmull-Clark • Doo-Sabin • Loop • Butterfly

  12. Catmull-Clark Subdivision

  13. Loop Subdivision

  14. Curved surface patches • Hermite curve & patches • Bezier curve & patches • Catmull-Rom spline • B-spline

  15. Hermite Curve • Given: starting and ending points and tangents • Determine cubic equation that satisfies constraints

  16. Hermite Patch • Given: starting and ending points and tangents • Determine bicubic equation that satisfies constraints

  17. Fractal geometry • Objects that exhibit similarity under scale • e.g., Koch curve • Mathematically curious - continuous, infinite length, nowhere differentiable, bounded

  18. Serpinski Gaskets

  19. Other fractal shapes Mandelbrot set Coastlines Mountains statistically self-similar under scale

  20. Create mountains • Subdivide triangle • Displace upward • Reduce amplitude of displacement • Recurse on subdivisions

  21. Implicit functions & isosurfaces • f(x,y,z) = 0 • “density” function • distance-based implicit functions • Marching Cubes algorithm

  22. Marching Cubes :convert to polygons

  23. Implicit functions & isosurfaces

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