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Chapter 11

Chapter 11. Curves. Modeling. Define an object or group of objects in terms of its form or shape surface modeling solid modeling particle system modeling. Surface Model. No interior information Surface only Visible Man Fly Thru Bill Lorenson, GE. Solid Model.

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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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