L-systems

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NUBS by Brian Wyvill
Whats that?
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Uniform B-Splines
Basis functions for B-Splines. B-splines
approximate a series of m1 Control points. (P0,
P1 .. Pm) M ? 3 M 2 cubic curve segments (Q3,
Q4 .. Qm) Parameter range for Qi is ti ? t ?
ti1 There is a knot between Qi and Qi1 at ti
tiknot value. (m-1 knots)
t30 to t41 to t52 etc. Qm defined by points.
(Pm-3, Pm-2, Pm-1, Pm ) over parameter range
tmm-3 to tm1m-2
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Uniform B-Splines
Geometry Vector for B-Splines.
Pi-3 Pi-2 Pi-1 Pi
GBs
, 3 ? i ? m
T (t - ti)3 (t - ti)2 (t - ti)1 1 Qi
(t) Ti MB GBs , ti ? t ? ti1
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Non-Uniform B-Splines
Effect of Multiple Knots
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NUBS
Piecewise continuous curve approximating control
points P0 to Pm Knot Value sequence is
non-decreasing sequence of knot values t0 through
tm4 ie there are 4 more knots than control
points.
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NUBS
Qi (t) Pi-3 Bi-3,4 (t) Pi-2 Bi-2,4
(t) Pi-1 Bi-1,4 (t) Pi Bi-4 (t) 3 ? i ?
m ti ? t lt ti1 Bi,1 (t) 1, ti ? t lt ti1
otherwise zero. Bi,2 (t) (t ti) /(ti1
ti)Bi,1(t) (ti2 t) /(ti2 ti1)Bi1,1 (t)
Bi,3 (t) Bi,4 (t) B0,1 (t) B1,1 (t) B2,1
(t) 0 B3,1 (t)1 then 0 for igt3 B0,2 (t)
something B0,1 (t) something B1,1 (t)
0 B1,2 (t) something B1,1 (t) something
B2,1 (t) 0 B2,2 (t) something B2,1 (t) (t4
t) /(t4 t3) B3,1 (t) (1-t)1 This is 1
when t0 and linearly decreasing until 0 at t1
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The Haar Wavelet
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