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Chapter 11

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  1. Chapter 11 Curves

  2. Modeling • Define an object or group of objects in terms of its form or shape • surface modeling • solid modeling • particle system modeling

  3. Surface Model • No interior information • Surface only • Visible Man Fly Thru Bill Lorenson, GE

  4. Solid Model • Interior information is preserved. • Requires huge amounts of space.

  5. Particle-system Model • Model objects consisting of many of the same shape. • Water, fire, trees, dandelion seeds, bubbles and grapes might be modeled by particle systems.

  6. Polygonal Modeling z • 0 dimension • point • 1 dimensions, length (x) • line • 2 dimensions, length (x) and height (y) • plane • 3 dimensions, length (x), height (y) and width (z) • cube x y

  7. Polygonal Modeling • Vertex • Polygon • Polygonal Approximation

  8. Polygon Reduction (or Polygon Culling or Polygon Thinning) • Specifying maximum number of polygons • Specify minimum angle between polygons

  9. Another Global Polygon Adjustments • Polygon expansion • the opposite of polygon reduction

  10. Local Operations • Restricting polygon expansion to the areas of high detail • efficient use of polygons

  11. Linear Approximation of a Curve • 2 or 3 dimensions • linear approximation (polyline) • simple concept • awkward to edit • many points necessary for good approximation of curve • never smooth

  12. Parameterized Curves • Known as spline curves • have direction • beginning point • ending point • may be non planar

  13. Explicit Functions • y = f(x) • x = g(y) • y = mx + b • Problems -- with vertical lines and circles • y = sqrt(r^2 - x^2) • y = -sqrt(r^2 - x^2) • Only if 0<=|x|<=r • Problems compound with 3D • z = f(x,y) can’t work because a given x,y generates multiple points on a sphere.

  14. Curves • Developed in the CAD industry • aviation • automobiles • Hermite Curve • Bezier Curve

  15. Splines • Shipbuilders forced wood splines around “ducks” • Curve shaped by • control points • control vertices • Types of Splines used in Computer Graphics • natural spline • B-Splines • NURBs

  16. Continuity • C0 - connectedness • C1 - smoothness (no tangent breaks) • C2 - curvature

  17. Hermite Curve • Used for interpolation of keyframe data • Use • hermite basis functions • points p1 and p2 and tangent vectors t1 and t2 t1 p1 p2 t2

  18. Bezier Curve • Developed at Renault by Pierre Bezier • Pair of endpoints and control points (not on the curve) Control Point Control Point Endpoint Endpoint

  19. B Spline • B stands for basis or blending function • Control points do not interpolate the curve. • Difficult to edit because control points must be moved significantly to see change in curve shape. • Can only be cut at a knot (between curve segments -- difficult to ascertain since control points are not on the curve)

  20. NURBSNon-Uniform Rational B-Spline • There is no NURB, only NURBS • a type of B-spline • each control point can have a weight • can cut anywhere on the length of the curve

  21. Spline Patches • Original direction of curve, u • Moved through space along a second curve with direction v

  22. Homework for next week • Work Day Wednesday 4/11. No class. • Read Lasseter, John. “Principles of Traditional Animation Applied to 3D Computer Animation”, Proceedings of SIGGraph 1987.

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