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

Procedurally generated

Controlled randomness

sweep operator
Sweep Operator

Combines extrusion, lofting, and surface of revolution

more sophisticated approaches
More sophisticated approaches
  • CSG
  • Subdivision surfaces
  • Curves & surfaces
  • Fractals
  • Implicit functions
csg constructive solid geometry
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
subdivision surfaces
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
subdivision surfaces1
Subdivision surfaces
  • Round corners
  • Catmull-Clark
  • Doo-Sabin
  • Loop
  • Butterfly
curved surface patches
Curved surface patches
  • Hermite curve & patches
  • Bezier curve & patches
  • Catmull-Rom spline
  • B-spline
hermite curve
Hermite Curve
  • Given: starting and ending points and tangents
  • Determine cubic equation that satisfies constraints
hermite patch
Hermite Patch
  • Given: starting and ending points and tangents
  • Determine bicubic equation that satisfies constraints
fractal geometry
Fractal geometry
  • Objects that exhibit similarity under scale
  • e.g., Koch curve
  • Mathematically curious - continuous, infinite length, nowhere differentiable, bounded
other fractal shapes
Other fractal shapes

Mandelbrot set

Coastlines

Mountains

statistically self-similar under scale

create mountains
Create mountains
  • Subdivide triangle
  • Displace upward
  • Reduce amplitude of displacement
  • Recurse on subdivisions
implicit functions isosurfaces
Implicit functions & isosurfaces
  • f(x,y,z) = 0
  • “density” function
  • distance-based implicit functions
  • Marching Cubes algorithm