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CS 445/645 Fall 2001

CS 445/645 Fall 2001. Splines/Film/Animation. Final Exam. Thursday, December 13 th from 7 – 10 p.m. Room Olsson 011 You may use one sheet of notes (8.5 x 11’’) You can write on both sides of paper Not everything from semester will be on test

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CS 445/645 Fall 2001

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  1. CS 445/645Fall 2001 Splines/Film/Animation

  2. Final Exam • Thursday, December 13th from 7 – 10 p.m. • Room Olsson 011 • You may use one sheet of notes (8.5 x 11’’) • You can write on both sides of paper • Not everything from semester will be on test • Class on Thursday will outline material and review

  3. Wanted: Undergrad TAs • Undergrads needed to TA and grade for CS courses • TA labs for CS101, CS201, CS216, etc. • Office hours, grading for all undergrad courses • To apply, fill out form at: • http://www.cs.virginia.edu/jobs • Get to know CS professors (letters of reco) • Good for you resume • You folks play an important role in delivering high-quality CS courses at UVA

  4. Teaching Evaluations • Would you like this course to continue to be taught? • We need feedback that students like this course and want it to be a permanent part of our course offerings • Think about what you liked in the course • And what you didn’t like • And fill out a survey

  5. Today • One more type of curve to look at • B-splines • Some leftover slides about films • Some slides about animation

  6. Specifying Curves • Control Points • A set of points that influence the curve’s shape • Knots • Control points that lie on the curve • Interpolating Splines • Curves that pass through the control points (knots) • Approximating Splines • Control points merely influence shape

  7. Parametric Cubic Splines • Can represent this as a matrix too

  8. Hermite Cubic Splines • An example of knot and continuity constraints

  9. Resulting Hermite Spline Equation

  10. Demonstration • Hermite

  11. Blending Functions • By multiplying first two components, you have four functions of ‘t’ that blend the four control parameters

  12. Hermite Blending Functions • If you plot the blending functions on the parameter ‘t’

  13. Bézier Curves • Similar to Hermite, but more intuitive definition of endpoint derivatives • Four control points, two of which are knots

  14. Bézier Curves • Similar to Hermite, but more intuitive definition of endpoint derivatives • Four control points, two of which are knots

  15. Bézier Curves • The derivative values of the Bezier Curve at the knots are dependent on the adjacent points • The scalar 3 was selected just for this curve

  16. Bézier Basis and Geometry Matrices • Matrix Form • But why is MBezier a good basis matrix?

  17. Bézier Blending Functions • Look at the blending functions • This family of polynomials is calledorder-3 Bernstein Polynomials • They are all positive in interval [0,1] • Their sum is equal to 1

  18. Bézier Blending Functions • Thus, every point on curve is linear combination of the control points • The weights of the combination are all positive • The sum of the weights is 1 • Therefore, the curve is a convex combination of the control points

  19. Bézier Curves • Bezier

  20. Splines - History • Draftsman use ‘ducks’ and strips of wood (splines) to draw curves • Wood splines have second-order continuity • And pass through the control points A Duck (weight) Ducks trace out curve

  21. Why more spline lectures? • Bezier and Hermite splines have global influence • Moving one control point affects the entire curve • B-splines consist of curve segments whose polynomial coefficients depend on just a few control points • Local control

  22. Uniform B-Splines • Approximating Splines • Approximates m+1 control points • P0, P1, …, Pm, m ¸ 3 • Curve consists of m –2 cubic polynomial segments • Q3, Q4, … Qm • t varies along B-spline as Qi: ti <= t < ti+1 • ti (i = integer) are knot points that join segment Qi-1 to Qi • Curve is uniform because knots are spaced at equal intervals of parameter,t

  23. Uniform B-Splines • First curve segment, Q3, is defined by first four control points • Last curve segment, Qm, is defined by last four control points, Pm-3, Pm-2, Pm-1, Pm • Each control point affects four curve segments

  24. B-spline Basis Matrix • Formulate 16 equations to solve the 16 unknowns • The 16 equations enforce the C0, C1, and C2 continuity between adjoining segments, Q

  25. B-Spline • Points along B-Spline are computed just as with Bezier Curves

  26. B-Spline • By far the most popular spline used • C0, C1, and C2 continuous

  27. B-Spline • Locality of points

  28. Nonuniform, Rational B-Splines(NURBS) • The native geometry element in Maya • Models are composed of surfaces defined by NURBS, not polygons • NURBS are smooth • NURBS require effort to make non-smooth

  29. What is a NURB? • Nonuniform: The amount of parameter, t, that is used to model each curve segment varies • Rational: The parametric functions that define each curve segment are ratios of polynomals:

  30. What do we get? • Nonuniformity permits either C2, C1, or C0 continuity at join points between curve segments • Nonuniformity permits control points to be added to middle of curve • Rational curves are invariant under rotation, scaling, translation, and perspective transformations of the control points (nonrational curves are not preserved under perspective projection) • This means you can transform the control points and redraw the curve using the transformed points • If this weren’t true you’d have to sample curve to many points and transform each point individually

  31. What do we get? • NURBS represent conics

  32. Converting Between Splines • Consider two spline basis formulations for two spline types

  33. Converting Between Splines • We can transform the control points from one spline basis to another

  34. Converting Between Splines • With this conversion, we can convert a B-Spline into a Bezier Spline • Bezier Splines are easy to render

  35. Rendering Splines • Horner’s Method • Incremental (Forward Difference) Method • Subdivision Methods

  36. Horner’s Method • Three multiplications • Three additions

  37. Forward Difference • But this still is expensive to compute

  38. Forward Difference • So, at startup compute these things • Successive values of x requires 3 additions

  39. Subdivision Methods • Bezier

  40. Rendering Bezier Spline public void spline(ControlPoint p0, ControlPoint p1, ControlPoint p2, ControlPoint p3, int pix) { float len = ControlPoint.dist(p0,p1) + ControlPoint.dist(p1,p2) + ControlPoint.dist(p2,p3); float chord = ControlPoint.dist(p0,p3); if (Math.abs(len - chord) < 0.25f) return; fatPixel(pix, p0.x, p0.y); ControlPoint p11 = ControlPoint.midpoint(p0, p1); ControlPoint tmp = ControlPoint.midpoint(p1, p2); ControlPoint p12 = ControlPoint.midpoint(p11, tmp); ControlPoint p22 = ControlPoint.midpoint(p2, p3); ControlPoint p21 = ControlPoint.midpoint(p22, tmp); ControlPoint p20 = ControlPoint.midpoint(p12, p21); spline(p20, p12, p11, p0, pix); spline(p3, p22, p21, p20, pix); }

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