THINKING IN PYRAMIDS

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Angelina Gillian Phelps
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1 ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think in pyramids when ooking at B-spine curves. severa of the pyramids that are usefu in discussing B-spine curves. This reviews The B-Spine Curve Geometric Definition Given a set of Contro Points {P 0, P 1,..., P n }, an order k, and a set of knots {t 0, t 1,..., t n+k } the geometric definition of a B-Spine curve of order k is given by P = P (k 1) if t [t, t +1 ) (1) where P (j) i = (1 τ j i )P(j 1) i 1 + τ j i P(j 1) i if j > 0, P i if j = 0. (2) and τ j i = t t i t i+k j t i () Here, Equation (1) tes us which contro points to use, depending on where the vaue of t ands in the knot sequence. This insures a smooth transition between the Bézier segments that make up the curve. Equation (2) is the recursive definition of P (j) i that we must use, and Equation () shows us how to cacuate the vaue of τ j i, depending again on where t ies in the knot sequence.

2 There is a good pyramid structure that aows one to visuaize this process.. P k+1 k+2 P k+2 k+ k+ P k+ P k+ P (k 2) 1 P (k 1) P (k 2) P P 1 P. From Equation (2), we see that any P in this pyramid is cacuated as a convex combination of the two P functions immediatey to it s eft. This tes us that if t [t, t +1 ), then the cacuation of P = P (k 1) ony depends on the k contro points P k+1, P k+2,..., and P. As t moves aong the knot sequence, from t k 1 to t n+1, the contro points that affect P shift when different intervas [t i, t i+1 ] are entered. It may be usefu to iustrate a smaer exampe. Suppose we have six contro points P 0, P 1, P 2, P, P, and P, k = (so we wi have cubic segments), n =, and we have knots {t 0 t 1 t 2 t t t t 6 t 7 t 8 t 9 } = { }. Then t ranges between [t k 1, t n+1 ] = [t, t 6 ] = [0, ], and the curve has three Bézier segments: one when 0 t 1, one when 1 t 2, and one when 2 t. When t [0, 1], our 2

3 construction pyramid ooks ike (horizonta this time). P () P 0 P 1 P 2 P P P where ony P 0, P 1, P 2, and P are used in the cacuation of P = P () [You wi note that I eft off the quantity in the pyramid to save some space]. We note that P and P are not used in the cacuation and do not affect the curve when t [0, 1]. When t [1, 2], our construction pyramid ooks ike the foowing: P () 2 P 0 P 1 P 2 P P P where ony P 1, P 2, P, and P are used in the cacuation of P. Note that P 0 and P are not used in the cacuation and do not affect the curve when t [1, 2]. When t [2, ], our construction pyramid ooks ike the foowing: P () P 0 P 1 P 2 P P P where ony P 2, P, P, and P and used in the cacuation of P. Note that P 0 and P 1 are not used in the cacuation and do not affect the curve when t [2, ]. This gives a good indication of the oca contro that a B-spine curve gives. Ony a few contro points are used to cacuate P for any t, and this means that contro points outside of this cacuation range do not affect the curve in this area. Aso, if we examine the bending functions N i,k in the anaytic definition, we see that N i,k = t t i N i,k 1 + t i+k t N i+1,k 1 t i+k 1 t i t i+k t i+1

4 where 1 if t [t i, t i+1 ), N i,1 = 0 otherwise. If we draw the foowing pyramid where various N.,. functions in the pyramid are a weighted sum of the two N.,. functions immediatey to its right from the above equation we get N i,k N i,k 1 N i+1,k 1 N i,1 N i,2. N i+1,1 N i,k 2. N i+1,k 2 N i+2,k 2.. N i+k 2,1 N i+k 1,2 N i+k 1,1 We can see that N i,k (at the far eft) eventuay depends on N i,1, N i+1,1,..., and N i+k 1,1 (at the far right), and so we can concude that N i,k 0 ony if t [t i, t i+k ). [Because if t [t i, t i+k ), then N i,1, N i+1,1,..., and N i+k 1,1 woud a be zero). Aso, for each vaue of t, ony a imited number of the N i,k are non-zero. This can be seen by

5 organizing a different pyramid. If t [t, t k+1 ) then we can arrange the foowing pyramid. N,1 N,2 N 1,2 N,k N 1,k 1. N 1,k N,. N 1, N 2,.. N k+2,k N k+1,k 1 N k+1,k Here, the fact that N,1 0 impies that N,2 0 and N 1,2 0 (the two bending functions immediatey to its right in the pyramid) are aso non-zero immediatey to its right. And, in genera, if N i,j 0 impies that N i,j 1 0 and N i 1,j 1 0 in the pyramid. From this, we can concude that if t [t, t +1 ), then ony N k+1,k, N k+2,k,..., and N,k are non-zero. For exampe, N 1,k = N k,k = 0. So, if t [t, t +1 ), ony k of the N.,. functions are non-zero, and so ony k of the contro points P k+1, P k+2,..., and P affect the point P on the curve. This makes sense, as a B-spine curve is a piecewise poynomia curve, and ony a few contro points shoud affect each poynomia piece. These pyramid structures wi be hepfu in the discussions about the B-spine representation. A contents copyright (c) Computer Science Department, University of Caifornia, Davis A rights reserved.

On-Line Geometric Modeling Notes

On-Line Geometric Modeling Notes CUBIC BÉZIER CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview The Bézier curve representation