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Lecture on Splines, Cubic Curves, and Interpolation

This lecture covers various topics related to splines, cubic curves, and interpolation, including linear and Hermite interpolation, spline basis functions, Catmull-Rom splines, Bezier curves, and more. It also explores the applications of these techniques in areas like visualization, robotics, and realistic rendering and animation.

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Lecture on Splines, Cubic Curves, and Interpolation

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  1. Lecture 15 Splines, Cubic Curves, and Interpolation Friday, 27 February 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Readings: Chapters 7, 11, Foley et al. Reference: Chapter 9, 10.1, 10.6-10.13, 10.16-10.17, Hearn and Baker 2e Next Class: 12.1-12.5, Foley et al.

  2. Splines

  3. Linear Interpolation

  4. Hermite Interpolation

  5. Hermite Interpolation

  6. Hermite Basis Matrix

  7. Hermite Basis Functions

  8. Catmull-Rom Spline

  9. Bezier Curve

  10. Chaiken’s Algorithm

  11. Bezier Curves

  12. Properties

  13. Subdivision and Extrapolation

  14. Piecewise Curves

  15. Bernstein Polynomials

  16. Bernstein Polynomials

  17. Turn To A Partner Exercise [1] • In-Class Exercise: Turn to A Partner • CG projects • Applications to visualization • Robotics • Realistic rendering and animation • List one • Methodology (if any) your projects share in common • Obstacle (if any) your projects share in common • After-class exercise: think about common obstacles (send e-mail or post)

  18. Turn To A Partner Exercise [2] • Team 1: • Team 2:

  19. Turn To A Partner Exercise [3] • Team 3: • Team 4:

  20. Turn To A Partner Exercise [3] • Team 5: • Team 6: • Team 7:

  21. Bicubic Surfaces • Readings • Sections 11.3, 12.1-12.5, Foley et al • Optional reference: Chapter 9, 10.1, 10.6-10.13, 10.16-10.17, Hearn and Baker 2e • Quick Review: Properties of Cubic Curves and Splines • Splines: B- (UN, NUN, NURNURBS), Beta- (-), Catmull-Rom, Kochanek-Bartels • Uniformity, rationality, continuity, control point and polygon properties • Interpolating Cubic Curves and Surfaces • deCasteljau’s algorithm for curves • Bicubic surface interpolation (concluded) • 3D Graphics Data Structures • This time: boundary representations (akaB-reps) • Next time: spatial partitioning representations • Visible Surface Determination (VSD): Introduction • Role of graphics data structures in VSD (more later) • Graphics data structures in computational geometry • Next Lecture: Basics of Constructive Solid Geometry (Survey)

  22. Comparison of Cubic Curves • Hermite • Blend 4 functions; no CP; full interpolation; C1 and G1 with constraints; fast • Bézier • Convex CP; interpolate 2 of 4 control points; C1 and G1 with constraints; fastest • B-splines • Uniform, nonrational • Convex CP, 4 points each, no interpolation; C2 and G2; medium • Nonuniform, nonrational • Convex CP, 5 points each, "no interpolation”; “up to” C2 and G2; slow • Nonuniform, rational • Convex CP, 5 points each, "no interpolation”; rational; “up to” C2 and G2; slow • Beta Splines (-Splines) • Convex CP; 6 points to control curve (4 local points, 2 global); C1 and G2; medium • Catmull-Rom Splines • General CP; interpolate or approximate 4 points per CP; C1 and G1; medium • Kochanek-Bartels Splines • General CP; interpolate 7 points per CP; C1 and G1; medium

  23. P1 P2 P0 P3 Interpolating Curves [1]:Recursive Subdivision • Intuitive Idea • Given • Curve (Bézier or uniform B-spline) defined using control polygons (CPs) • 4 control points P0, P1, P2, P3 • Problem: can’t get quite the right curve shape (not enough control points) • Solutions: increase degree of polynomial segments OR add CPs • Technique: recursive subdivision algorithm • Add control points by splitting existing CP up recursively • Compute CPs for left curveL0, L1, L2, L3, right curveR0, R1, R2, R3 • Stop when variation (curve-to-control point distance) is low enough • Purpose: display curve OR allow new control points to be manipulated

  24. P1 P2 P0 P3 Interpolating Curves [2]:deCasteljau’s Algorithm • Recursive Subdivision Algorithm for Interpolation [deCasteljau, 1959] • Purpose: display curve OR allow new control points to be manipulated • Display: fast and cheap (see below) • Properties • Cheap: can implement using subdivision matrices (Equations 11.52, 11.53, FVD) • Fast: rapid convergence due to… • Variation-diminishing property • Monotonic convergence to curve • Holds for all splines with convex-hull CPs • When Does It Work? • Uniform splines (uniformly-spaced knots) • Q: Can we subdivide NURBS? • A: Yes, by adding knots (expensive) [Böhm, 1980; Cohen et al, 1980] • Alternative approach: hierarchical B-splines [Forsey and Bartels, 1988]

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