1 Geometric Interpretation: Cubic Why Polar Forms? Why Polar - PDF document

Outline of Unit Outline of Unit Foundations of Computer Graphics Foundations of Computer Graphics (Spring 2012) (Spring 2012)  Bezier curves (last time) CS 184, Lecture 13: Curves 2  deCasteljau algorithm, explicit, matrix (last time) http://inst.eecs.berkeley.edu/~cs184  Polar form labeling (blossoms)  B-spline curves  Not well covered in textbooks (especially as taught here). Main reference will be lecture notes. If you do want a printed ref, handouts from CAGD, Seidel Idea of Blossoms/Polar Forms Idea of Blossoms/Polar Forms Idea of Blossoms/Polar Forms Idea of Blossoms/Polar Forms  (Optional) Labeling trick for control points and intermediate  Points on curve simply have u 1 =u 2 so that F(u) = f(u,u) deCasteljau points that makes thing intuitive  f is symmetric f(0,1) = f(1,0)  E.g. quadratic Bezier curve F(u)  Only interpolate linearly between points with one arg different  Define auxiliary function f(u 1 ,u 2 ) [number of args = degree]  f(0,u) = (1-u) f(0,0) + u f(0,1) Here, interpolate f(0,0) and f(0,1)=f(1,0)  Points on curve simply have u 1 =u 2 so that F(u) = f(u,u)  And we can label control points and deCasteljau points not on curve with appropriate values of (u 1 ,u 2 ) f(0,1)=f(1,0) f(0,1)=f(1,0) 00 01 11 1-u u 1-u u 0u 1u 1-u u f(u,u) = F(u) f(u,u) = F(u) uu F(u) = f(uu) = (1-u) 2 P0 + 2u(1-u) P1 + u 2 P2 f(0,0) = F(0) f(1,1) = F(1) f(0,0) = F(0) f(1,1) = F(1) Geometric interpretation: Quadratic Polar Forms: Cubic Bezier Curve Polar Forms: Cubic Bezier Curve Geometric interpretation: Quadratic 01=10 001 011 u 1u 1-u 111 000 1-u u 000 001 011 111 1-u u 1-u u 1-u u uu 0u 00u 01u 11u u 1-u u 1-u u 0uu 1uu 00 11 u 1-u uuu 1

Geometric Interpretation: Cubic Why Polar Forms? Why Polar Forms?  Simple mnemonic: which points to interpolate and 001 0u1 011 how in deCasteljau algorithm uu1  Easy to see how to subdivide Bezier curve (next) u11 which is useful for drawing recursively uuu 0uu  Generalizes to arbitrary spline curves (just label control points correctly instead of 00 01 11 for Bezier) 00u  Easy for many analyses (beyond scope of course) 000 111 Geometrically Subdividing Bezier Curves Subdividing Bezier Curves Drawing: Subdivide into halves (u = ½) Demo: hw4.exe  Recursively draw each piece 001 0½1 011  At some tolerance, draw control polygon  Trivial for Bezier curves (from deCasteljau algorithm): hence ½½½ widely used for drawing 0½½ ½½1 000 001 011 111 00½ ½11 000 00u 0uu uuu uuu uu1 u11 111 Why specific labels/ control points on left/right?  How do they follow from deCasteljau? 000 111 Geometrically Subdivision in deCasteljau deCasteljau diagram diagram Subdivision in 001 011 001 0½1 011 111 ½½½ 000 These (interior) points don’t 0½½ ½½1 appear in subdivided curves at all 000 001 011 111 00½ ½11 1-u u 1-u u 1-u u 00u 01u 11u Right part of Bezier curve Left part of Bezier curve 1-u u 1-u u (000, 00u, 0uu, uuu) (uuu, 1uu, 11u, 111) 0uu 1uu Always right edge of Always left edge of deCasteljau pyramid deCasteljau pyramid u 1-u 000 111 uuu 2

Summary for HW 2 Summary for HW 2 DeCasteljau DeCasteljau: Recursive Subdivision : Recursive Subdivision  Bezier2 (Bezier discussed last time)  Given arbitrary degree Bezier curve, recursively subdivide for some levels, then draw control polygon hw4.exe  Generate deCasteljau diagram; recursively call a routine with left edge and right edge of this diagram  You are given some code structure; you essentially just  DeCasteljau (from last lecture) for midpoint need to compute appropriate control points for left, right  Followed by recursive calls using left, right parts Outline of Unit Bezier: Disadvantages Outline of Unit Bezier: Disadvantages  Single piece, no local control (move a control point, whole curve changes) hw4.exe  Bezier curves (last time)  Complex shapes: can be very high degree, difficult  deCasteljau algorithm, explicit, matrix (last time)  In practice, combine many Bezier curve segments  Polar form labeling (blossoms)  But only position continuous at join since Bezier curves  B-spline curves interpolate end-points (which match at segment boundaries)  Unpleasant derivative (slope) discontinuities at end-points  Can you see why this is an issue?  Not well covered in textbooks (especially as taught here). Main reference will be lecture notes. If you do want a printed ref, handouts from CAGD, Seidel B B- -Splines Splines Polar Forms: Cubic Polar Forms: Cubic Bspline Bspline Curve Curve  Cubic B-splines have C 2 continuity, local control  Labeling little different from in Bezier curve  4 segments / control point, 4 control  No interpolation of end-points like in Bezier points/segment  Advantage of polar forms: easy to generalize  Knots where two segments join: Knotvector  Knotvector uniform/non-uniform (we only consider –1 0 1 0 1 2 uniform cubic B-splines, not general NURBS) Uniform knot vector: -2, -1, 0, 1, 2 ,3 Knot: C 2 continuity Demo: hw4.exe Labels correspond to this deBoor points 1 2 3 -2 –1 0 3

deCasteljau: Cubic B : Cubic B- -Splines Splines deCasteljau: Cubic B : Cubic B- -Splines Splines deCasteljau deCasteljau  Easy to generalize using –1 0 1  Easy to generalize using –1 0 1 0 1 2 0 1 2 polar-form labels polar-form labels  Impossible remember  Impossible remember 1 2 3 1 2 3 -2 –1 0 -2 –1 0 without without -2 -1 0 -1 0 1 0 1 2 1 2 3 -2 -1 0 -1 0 1 0 1 2 1 2 3 ? ? ? ? ? ? (2+u)/3 (1-u)/3 (1+u)/3 (2-u)/3 u/3 1-u/3 -1 0 u 0 1 u -1 0 u 0 1 u 1 2 u 1 2 u ? ? ? ? 0 u u 1 u u deCasteljau: Cubic B : Cubic B- -Splines Splines deCasteljau Explicit Formula (derive as exercise) Explicit Formula (derive as exercise)   P 0      Easy to generalize using 1 3 3 1 –1 0 1 0 1 2    1    M M P 1 3 6 3 0 polar-form labels      3 2 ( ) [ 1] F u u u u      6 3 0 3 0 2 P    Impossible remember   1 2 3  1 4 1 0  -2 –1 0  3  P without P0 P1 P2 P3 -2 -1 0 0 1 2 1 2 3 -1 0 1 -2 -1 0 -1 0 1 0 1 2 1 2 3 (2+u)/3 (1-u)/3 (1-u)/3 (2+u)/3 (1+u)/3 (1+u)/3 (2-u)/3 (2-u)/3 1-u/3 u/3 u/3 1-u/3 -1 0 u 0 1 u 1 2 u -1 0 u 0 1 u 1 2 u (1+u)/2 (1-u)/2 (1-u)/2 (1+u)/2 1-u/2 u/2 1-u/2 u/2 0 u u 1 u u 0 u u 1 u u 1-u u 1-u u u u u u u u Summary of HW 2 Summary of HW 2  BSpline Demo hw4.exe  Arbitrary number of control points / segments  Do nothing till 4 control points (see demo)  Number of segments = # cpts – 3  Segment A will have control pts A,A+1,A+2,A+3  Evaluate Bspline for each segment using 4 control points (at some number of locations, connect lines)  Use either deCasteljau algorithm (like Bezier) or explicit form [matrix formula on previous slide]  Questions? 4