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Procedural Data Generation: Techniques for 3D Models and Fractal Geometry

Explore advanced procedural modeling techniques for generating complex data and 3D models. This guide covers parameterized procedures, data amplification, and basic approaches such as extrusion, surface revolution, and lofting. Delve into more sophisticated methods like Constructive Solid Geometry (CSG), subdivision surfaces, and fractals. Learn how to represent shapes with implicit functions and apply algorithms like Marching Cubes to convert isosurfaces to polygons. Ideal for artists and developers aiming to master procedural techniques in graphics rendering.

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Procedural Data Generation: Techniques for 3D Models and Fractal Geometry

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Presentation Transcript

  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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