To Do To Do Computer Graphics (Fall 2005) Computer Graphics (Fall - PDF document

To Do To Do Computer Graphics (Fall 2005) Computer Graphics (Fall 2005) � Start on HW 2 (cannot be done at last moment) This (and previous) lecture should have all information need COMS 4160, Lecture 7: Curves 2 http://www.cs.columbia.edu/~cs4160 � Start thinking about partners for HW 3 and HW 4 � Remember though, that HW2 is done individually � Your submission of HW 2 must include partner for HW 3 Outline of Unit Outline of Unit Idea of Blossoms/Polar Forms Idea of Blossoms/Polar Forms � (Optional) Labeling trick for control points and intermediate deCasteljau points that makes thing intuitive � Bezier curves (last time) � E.g. quadratic Bezier curve F(u) � deCasteljau algorithm, explicit, matrix (last time) � Define auxiliary function f(u 1 ,u 2 ) [number of args = degree] � Points on curve simply have u 1 =u 2 so that F(u) = f(u,u) � Polar form labeling (blossoms) � And we can label control points and deCasteljau points not � B-spline curves on curve with appropriate values of (u 1 ,u 2 ) f(0,1)=f(1,0) � Not well covered in textbooks (especially as taught here). Main reference will be lecture notes. If you do f(u,u) = F(u) want a printed ref, handouts from CAGD, Seidel f(0,0) = F(0) f(1,1) = F(1) Idea of Blossoms/Polar Forms Idea of Blossoms/Polar Forms Geometric interpretation: Quadratic Geometric interpretation: Quadratic 01=10 � Points on curve simply have u 1 =u 2 so that F(u) = f(u,u) u � f is symmetric f(0,1) = f(1,0) 1u � Only interpolate linearly between points with one arg different 1-u � f(0,u) = (1-u) f(0,0) + u f(0,1) Here, interpolate f(0,0) and f(0,1)=f(1,0) 1-u u uu f(0,1)=f(1,0) 00 01 11 0u u 1-u u 1-u u 0u 1u 1-u u 00 11 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)

Geometric Interpretation: Cubic Polar Forms: Cubic Bezier Curve Polar Forms: Cubic Bezier Curve 001 011 001 0u1 011 uu1 111 u11 000 000 001 011 111 uuu 1-u u 1-u u u 1-u 0uu 00u 01u 11u 00u 1-u u 1-u u 0uu 1uu u 1-u 000 111 uuu Why Polar Forms? Why Polar Forms? Subdividing Bezier Curves Subdividing Bezier Curves � Simple mnemonic: which points to interpolate and how in Drawing: Subdivide into halves (u = ½) Demo: hw2.exe deCasteljau algorithm � Recursively draw each piece � At some tolerance, draw control polygon � Trivial for Bezier curves (from deCasteljau algorithm): hence widely used for drawing � Easy to see how to subdivide Bezier curve (next) which is useful for drawing recursively 000 001 011 111 000 00u 0uu uuu uuu uu1 u11 111 � Generalizes to arbitrary spline curves (just label control points correctly instead of 00 01 11 for Bezier) Why specific labels/ control points on left/right? � How do they follow from deCasteljau? � Easy for many analyses (beyond scope of course) Geometrically Geometrically 001 0½1 011 001 0½1 011 ½½½ ½½½ 0½½ 0½½ ½½1 ½½1 00½ 00½ ½11 ½11 000 000 111 111

Summary for HW 2 Subdivision in deCasteljau Subdivision in deCasteljau diagram diagram Summary for HW 2 001 011 � Bezier2 (Bezier discussed last time) � Given arbitrary degree Bezier curve, recursively subdivide 111 000 These (interior) points don’t for some levels, then draw control polygon hw2.exe appear in subdivided curves at all � Generate deCasteljau diagram; recursively call a routine 000 001 011 111 with left edge and right edge of this diagram 1-u u 1-u u u 1-u 00u 01u 11u � You are given some code structure; you essentially just Left part of Bezier curve Right part of Bezier curve need to compute appropriate control points for left, right 1-u u 1-u u (uuu, 1uu, 11u, 111) (000, 00u, 0uu, uuu) 0uu 1uu Always left edge of Always right edge of deCasteljau pyramid deCasteljau pyramid u 1-u uuu DeCasteljau: Recursive Subdivision : Recursive Subdivision Outline of Unit Outline of Unit DeCasteljau � Bezier curves (last time) � deCasteljau algorithm, explicit, matrix (last time) � Polar form labeling (blossoms) � B-spline curves � Not well covered in textbooks (especially as taught � DeCasteljau (from last lecture) for midpoint here). Main reference will be lecture notes. If you do � Followed by recursive calls using left, right parts want a printed ref, handouts from CAGD, Seidel Bezier: Disadvantages Bezier: Disadvantages B- B -Splines Splines � Cubic B-splines have C 2 continuity, local control � Single piece, no local control (move a control point, whole curve changes) hw2.exe � 4 segments / control point, 4 control points/segment � Complex shapes: can be very high degree, difficult � Knots where two segments join: Knotvector � In practice, combine many Bezier curve segments � Knotvector uniform/non-uniform (we only consider � But only position continuous at join since Bezier curves uniform cubic B-splines, not general NURBS) interpolate end-points (which match at segment boundaries) � Unpleasant derivative (slope) discontinuities at end-points Knot: C 2 continuity Demo: hw2.exe � Can you see why this is an issue? deBoor points

Polar Forms: Cubic Bspline Bspline Curve Curve deCasteljau: Cubic B : Cubic B- -Splines Splines Polar Forms: Cubic deCasteljau � Labeling little different from in Bezier curve � Easy to generalize using –1 0 1 0 1 2 polar-form labels � No interpolation of end-points like in Bezier � Impossible remember 1 2 3 -2 –1 0 � Advantage of polar forms: easy to generalize without –1 0 1 0 1 2 -2 -1 0 -1 0 1 0 1 2 1 2 3 ? ? ? ? ? ? Uniform knot vector: -2, -1, 0, 1, 2 ,3 -1 0 u 0 1 u 1 2 u Labels correspond to this 1 2 3 -2 –1 0 deCasteljau: Cubic B deCasteljau : Cubic B- -Splines Splines deCasteljau: Cubic B deCasteljau : Cubic B- -Splines Splines � 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 (2+u)/3 (1-u)/3 (1-u)/3 (1+u)/3 (1+u)/3 (2-u)/3 (2-u)/3 1-u/3 u/3 1-u/3 u/3 -1 0 u 0 1 u -1 0 u 0 1 u 1 2 u 1 2 u (1+u)/2 (1-u)/2 ? ? ? ? 1-u/2 u/2 0 u u 1 u u 0 u u 1 u u 1-u u u u u Summary of HW 2 Summary of HW 2 Explicit Formula (derive as exercise) Explicit Formula (derive as exercise)  P 0  − − � BSpline Demo hw2.exe  1 3 3 1  1   =   M M P 1 − 3 6 3 0     = 3 2 � Arbitrary number of control points / segments F u ( ) [ u u u 1] 6  −    3 0 3 0 P 2 � Do nothing till 4 control points (see demo)     1 4 1 0   � Number of segments = # cpts – 3 P 3   P0 P1 P2 � Segment A will have control pts A,A+1,A+2,A+3 P3 -2 -1 0 1 2 3 -1 0 1 0 1 2 � Evaluate Bspline for each segment using 4 control (1-u)/3 (2+u)/3 points (at some number of locations, connect lines) (1+u)/3 (2-u)/3 u/3 1-u/3 -1 0 u 0 1 u 1 2 u � Use either deCasteljau algorithm (like Bezier) or (1-u)/2 (1+u)/2 explicit form [matrix formula on previous slide] 1-u/2 u/2 0 u u 1 u u � Questions? 1-u u u u u