1. Bspline Notes Jordan Smith UC Berkeley CS184

2. Outline Bézier Basis Polynomials –Linear –Quadratic –Cubic Uniform Bspline Basis Polynomials –Linear –Quadratic –Cubic Uniform Bsplines from Convolution

3. Review of Bézier Curves DeCastlejau Algorithm V 001 V 111 V 000 V 011 Insert at t = ¾

4. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 Insert at t = ¾

5. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾11 0¾1 Insert at t = ¾

6. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾¾1 ¾11 0¾10¾1 0¾¾ Insert at t = ¾

7. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾¾1 ¾11 0¾1 0¾¾ ¾¾¾ Insert at t = ¾

8. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾¾1 ¾11 0¾1 0¾¾ ¾¾¾ Insert at t = ¾

9. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾¾1 ¾11 0¾1 0¾¾ ¾¾¾ Insert at t = ¾

10. Review of Bézier Curves DeCastlejau Algorithm 001 000 111 011 00¾ ¾¾1 ¾11 0¾1 0¾¾ ¾¾¾ Insert at t = ¾

11. Bézier Curves Summary DeCastlejau algorithm –Evaluate Position(t) and Tangent(t) –Subdivides the curve into 2 subcurves with independent control polygons Subdivision of Bézier curves and convex hull property allows for: –Adaptive rendering based on a flatness criterion –Adaptive collision detection using line segment tests

12. Linear Bézier Basis Poly’s Bez 1 (t) = VtVt V0V0 V1V1 V0V0 VtVt V1V1 = (1-t) V 0 + t V 1 1-tt Knots: 012

13. Quadratic Bézier Basis Poly’s V 01 V tt V0tV0t Vt1Vt1 V 00 V 01 V 11 1-tt t t Bez 2 (t) == (1-t) 2 V 00 + 2(1-t)t V 01 + t 2 V 11 V 00 V0tV0t V 11 Vt1Vt1 V tt

14. Knots: Quadratic Bézier Basis Poly’s Bez 2 (t) = (1-t) 2 V 00 + 2(1-t)t V 01 + t 2 V 11 012

15. Cubic Bézier Basis Poly’s 001 111 011 00t tt1 t11 0t10t1 0tt ttt V ttt V 0tt V tt1 V 00t V0t1V0t1 V t11 V 111 V 011 V 001 V 000 1-tt t t t t t Bez 3 (t) == (1-t) 3 V 000 + 3(1-t) 2 t V 001 + 3(1-t)t 2 V 011 + t 3 V 111 000

16. Knots: Cubic Bézier Basis Poly’s Bez 3 (t) = (1-t) 3 V 000 + 3(1-t) 2 t V 001 + 3(1-t)t 2 V 011 + t 3 V 111 012

17. Blossoming of Bsplines 234 123 456 345 01765432 Knots:

18. Blossoming of Bsplines 234 123 456 345 233.53.545 33.54 017654323.5 Knots:

19. Blossoming of Bsplines 234 123 456 345 233.53.545 33.54 33.53.53.53.54 017654323.5 Knots:

20. Blossoming of Bsplines 234 123 456 345 233.53.545 33.54 33.53.53.53.54 017654323.5 3.53.53.5 Knots:

21. Bspline Blossoming Summary Blossoming of Bsplines is a generalization of the DeCastlejau algorithm Control point index triples on the same control line share 2 indices with each other Inserting a knot (t value) –Adds a new control point and curve segment –Adjusts other control points to form a control polygon Inserting the same t value reduces the parametric continuity of the curve A control point triple with all 3 indices equal is a point on the Bspline curve

22. Uniform Linear Bspline Basis Poly’s B 1 (t) = VtVt V0V0 V1V1 V0V0 VtVt V1V1 = (1-t) V 0 + t V 1 1-tt Knots: 012

23. Uniform Quadratic Bspline Basis Poly’s B 2 (t) = V tt V0tV0t Vt1Vt1 V -10 V 01 V 12 V 01 Vt1Vt1 V0tV0t V -10 V 12 V tt

24. Uniform Quadratic Bspline Basis Poly’s Knots: -20132 V -10 V 01 V 12

25. Uniform Cubic Bspline Basis Poly’s B 3 (t) = V 123 V 012 V -101 V -2-10 V ttt V 0tt V tt1 V -10t V0t1V0t1 V t12 -101 123 012 -10t tt1 t120t10t1 0tt ttt -2-10

26. Uniform Cubic Bspline Basis Poly’s Knots: -3-201432 V -2-10 V -101 V 012 V 123

27. Uniform Bsplines from Convolution 10102303214102 =  10  10  1023 = 102 = 10