Slide 129 October 1999CS Computer Graphics (Top Changwatchai) Bézier Curves - Results of Derivation Tangents at endpoints are equal to endpoint slopes of control graph Bézier curve lies within convex hull of control points, since each blending function is  0 and their sum is 1 Closed loop - first and last points at same position Weighting - multiple control points at same position give that position higher weight To avoid calculating high-degree polynomial, piece together smaller sections (cubic Bézier is common) –Automatically have C 0 continuity –Easy to get C 1 continuity: Each curve section has same number of control points Points adjacent to connecting points must be collinear and equally spaced –Can have C 2 continuity, but this can be unnecessarily restrictive

Slide 229 October 1999CS Computer Graphics (Top Changwatchai) Cubic Bézier Curves It is convenient to specify cubic Bézier curves See pp in Hearn and Baker for blending functions and basis matrix Later, if there is time, Dr. Chawla will show you how to convert between cubic basis matrices Bézier surfaces can be specified by Cartesian product of Bézier blending functions

Slide 329 October 1999CS Computer Graphics (Top Changwatchai) B-splines: Differences with Bézier See Section 10-9 of Hearn and Baker B-splines are the most widely used class of approximating splines Advantages over Bézier –Degree of B-spline can be set independent of the number of control points –Local control –(But remember, Bézier curves can be connected easily) Trade-off –More complex than Bézier Other –If cubic, has C 2 continuity –Does not necessarily interpolate endpoints

Slide 429 October 1999CS Computer Graphics (Top Changwatchai) B-spline Overview p0p0 pnpn u = 0u = 1 BEZ k,n Bézier global control u n+d u0u0 u d-1 udud u n+1 unun B 0,d B n,d local control Control point Knot Drawn curve Curve not drawn

Slide 529 October 1999CS Computer Graphics (Top Changwatchai) B-splines How to specify a general B-spline –n: number of control points = n+1 –d: degree of polynomial = d-12  d  n+1 –Control points (p 0,..., p n ) –Knot vector [ u 0... u n+d ] Features –Continuity over range of u is C d-2 –n+1 blending functions Each blending function B k,d is only defined for d subintervals, u k  u  u k+d –Local control Each section of spline curve is influenced by d control points Any one control point can affect the shape of at most d curve sections –Curve is drawn for u min  u  u max u min = u d-1, u max = u n+1 –B-splines are tightly bound to control points Each section lies within convex hull of at most d+1 control points –B-spline modification Can vary the number of control points without changing degree of polynomial, as long as ( 2  d  n+1 ) holds To pull curve closer to control-point position, specify that position multiple times u n+d u0u0 u d-1 udud u n+1 unun B 0,d B n,d

Slide 629 October 1999CS Computer Graphics (Top Changwatchai) B-splines B-splines are described according to knot-vector class: Uniform –Spacing between knot values is constant –Blending functions are periodic (all have same shape) –See Figure and Equation in textbook –Start and end positions of curve (umin and umax) are weighted averages of d-1 control points –To close curve, duplicate first d-1 control points at end of sequence –Cubic, periodic d = 4 Can derive boundary conditions, basis vectors, and blending functions Open uniform –Uniform except for endpoints –Knot values at each endpoint repeated d times –Characteristics very similar to Bézier curves When d = n+1, open B-splines reduce to Bézier splines Nonuniform –Unequal spacing –Reduces continuity