Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: The distance between a uniform (B-)Spline and its control polygon
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Title: The distance between a uniform (B-)Spline and its control polygon
Series Title: Department of Computer and Information Science and Engineering Technical Reports
Physical Description: Book
Language: English
Creator: Lutterkort, David
Peters, Jorg
Publisher: Department of Computer and Information Science and Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: September 10, 1998
Copyright Date: 1998
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Volume ID: VID00001
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Holding Location: University of Florida
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THE DISTANCE BETWEEN A UNIFORM (B-)SPLINE AND ITS
CONTROL POLYGON


DAVID LUTTERKORT AND JORG PETERS

ABSTRACT. We prove a sharp bound on the distance between a uniform .i;,i.
and its B- .i;,.. control polygon in terms of the second differences of the
control points. This bound decreases by a factor of 4 under uniform refinement.



1. UNIFORM B-SPLINES
A piecewise polynomial p of degree d in B-Spline form,

p(t) Z bkNd(t)
kEZ
is defined by a nondecreasing knot sequence (tk) and the control points (bk). Both
(tk) and (bk) are sequences of real numbers which we assume, for -i;!i1. i- to be
biinfinite. The knot sequence enters this equation through the well-known recursion
formulas for the B-Spline basis, see e.g. [dB93].
For .. '! ...... B-Splines, the knots are all equidistant and we 11i i- choose the knot
sequence tk = k without loss of _. i. i l.i .
The control "'v'., .... of p is the piecewise linear interpolant of the control points
bk attached to their Greville abscissae t*,
Ik+d
1 d+1
t = d i=k+ 2
i=k+l
Over the interval [t*, t*,+] the k-th piece fk of the control polygon I is given by
(t) = bk(t+l t) + bk+l(t tl) t E r'* tL+1l.
The centered second differences of b are defined as A2bk = bk-1 2bk + bk+l.
\\ oII this notation in place, we can state our main result:

Theorem 1.

(1.1) I--
24
where I'- i = "' 'i") (t)I and I_ I = I\ A2bk

The proof of the theorem is along the same lines as the proof of the corresponding
i;!. II L i- for polynomials in B6zier-form in [NPL98].

F!* ... We will prove the theorem by concentrating on each linear piece of the
control polygon separately and showing that, for ;,!i k, AI!- < I-_,1 I(d +

Date: September 10, 1998.
Technical Report TR-98-013, available from http://Awww.cise.ufl.edu/research/i. i ...i .











DAVID LUTTERKORT AND JORG PETERS


1)/24 over :,t*+l]. Once we have established that, it follows that |!- =
iin. I!|-' is also bounded by A_ I (d+ 1)/24.
We write p f over [t*, t+,] as Ei bi aki(t), where
t* -t i=k
k+lk
aki (t) = Nd (t) t -t i=k +l
0 otherwise.

O!1- finitely 111 ii- aoki have support on k+]. Which ones exactly depends on
the degree d and on k, but not on i. For -iHl~l! i- let f = f(d,k) and g = .,./ k)
be such that oki = 0 for i t E [t, t ] and all i V {f, f +1,...,g}.
The partition of It il' E N = 1 and the fact that t*+, t* = 1 imply that
Ei aki = 0. Similarly,

S-ai = t aki = +1 +1( ) t0* = o0
i i i

because of the linear precision of B-Splines, tN = t.
These two qualities together prove that the functions 3ki given by
i
/Okit) = E (i- j)aki

are all non-negative: since Zi (ki = i iaki = 0 we can express ki3 as

k (i )M3 i (j i) N i > k + 1.
The non-ii. ir of the 3ki now follows from the fact that all the '... int i 0 on
the right hand side and all the Nd are non-negative
Again, only finitely 11i i of the 3ki are non-zero. More specifically, we have
3ki = 0 for i < f since aki = 0 for these values of i; we also have that 3ki = 0 for
i > g since Ej=f(i j)akj = 0.
The 3 are second anti 1!I. i. i. of the a, in the sense that aki = A2Oki. Thus,
we can use summation by parts to rewrite p & through

p 4k = b aki = b A20ki A2bi Oki
i i i
and arrive over the interval [t;, t.;+] at the estimate

b,.- i, < Ah=
i i i
The theorem follows if we can show that q(t) := Oi ki < (d + 1)/24.
9 9 i 9 9 9--
O i Y f E A Ef j)a E E(' j)ai i E 5i
i i=f i=f j=f j=f i=j j i=0
z (~\i> y ((j kz )Nd
(j gj j











UNIFORM B-SPLINE BOUND


Ej (j2)N d is a quadratic polynomial which we express in monomial form using
the formulas in appendix A:
Ok t2d+2 1 1 ,
,ki = t2/2 (k + 2 )t+ (d + 2k)2 + (-' I + 13d + 10)

=1 ( 21 2 +
S (t* + t+l)t + t + d-2)

This shows that q is a non-negative quadratic polynomial with positive leading
*... th, ,. 1 It therefore attains its maximum over ', t,] at one of the endpoints
of the interval. Its values there are
d+l
q(t;) = (t;+1) = 24
24
and hence q(t) < (d + 1)/24 for all t E [t;, t;+, E
2. UNIFORM REFINEMENT
An important operation on B-Splines is the refinement of the knot sequence or
knot insertion. It changes the representation of the piecewise polynomial p over the
old knot sequence to one over an enlarged knot sequence. As a consequence, the
new control polygon b approximates p more closely than the old control polygon b.
For uniform B-Splines, we consider the refinement of the knot sequence tk = k
to the sequence tk = k/2, halving the distance between knots. We now have two
representations for p,
p(t)= bkNd(t k) = kh:- - k),
k k
where new control points bk are given by
d/2] /\d/2/
(2.2) b2i = 2-d 2 (2 d d bij = 2d d -l)b
J=0 2-1 + 2\j j=0 \ 2j
j=0 j=0
rd/2] Fd/2]
(2.3) 2i+1 = 2 d d jd b = 2 -d E /d+ bi-j.
j=0o j=o
Lemma 2. 1! second fI', .. A2b, of the new control ,.,'/.,'. are related to
the second 7I'fT .. .. A2bi of the old control ;I,,.l, ... through

2d A2b2i = ( d2j l 2bi-j
2j I

2 A22i+1 = (d 1) A2b,_.

j, .... We use the fact that the second [t!!i Abi and Abi are related to the
second derivative of p by
p"(t)= VbkNd-2(t k) = bkNdd-2(2t k)
k k
= Abk-1Nd-2(t k) = E Abk-1Nd-2(2t k).
k k











DAVID LUTTERKORT AND JORG PETERS


We have to be careful to change from the backward second 1!t [ i, i!! Vai = ai -
2ai1 +ai-2 to the centered second 'lit. i i- A A. They are related by Vai = Auai_.
Together with the refinement formulas (2.2) and (2.3) we get for even indices
[d-2]
Vb2i+2 = 2 -d (d ) vbi+l

[d/2] -1 d-
j-o
Ab2i+1 = 2- 2 1) Ab
j=0
and for the odd indices


Vb2i+1 = 2-d
j=o
[d/2] -
Ab2i = 2-d
j=o


(d +l)Vbi-j
(2j + 1i)


2j i) +1Vb,


2d ~ d 1)Abi-
j=1 (2 1


Corollary 3.
1
4
S...... This is a direct consequence of the fact that Tj (d 1)
-- ii!ii. 1i of the binomial ..ti. i! which imply that


S(d-21)
^ J


2d-1 and the


S(j = 2d-2
2j I


2.1. Special cases. For quadratic B-Splines, uniform refinement is ('I! ii,!:, al-
gorithm and
b2i = 2-2(, t + bi)
b2i+1 = 2-2(bi-, + ; 1.
This yields
1
A2b2i = A2b2i-1 = A2bi_
4
Similarly, for cubic B-Splines we have
62i = 2-3(bi-2 +-' Lt + bi)
b2i+1 = 2-3(4bi-1 + 4bi).
and
1 1
A2 A2i= A2bi-_ and A2b2i+1 = (A2bi_ + A2b).
4 8












UNIFORM B-SPLINE BOUND


APPENDIX A. REPRESENTING t2 AS A UNIFORM B-SPLINE
In general, monomials can be represented as B-Splines through1


A


E tk+l () Nh
(oEnd i=1


Here, IId is the set of all permutations of the numbers {1,..., d}.
For uniform B-Splines and j = 2, this boils down to:
2= I E E (tk+,(1)tk+,(2))Nd
2k a2ri

= ((k+o (1))(k + (2)))N
k UErid


2 E (k +P)(k + q)
k p=1 q=1

k 6
k


d
- (k+p)2) N
p=1

k(k + d + 1) + 2d2 +3d+1)] Nd


= E () (3(d + 2k)2 + 5d + 2 + 12k)NJ.


REFERENCES
[dB93] Carl de Boor. E. t. .ii.. basics. In Fundamental Developments of Computer Aided
Geometric Modeling, pages 27-49. Academic Press, London, 1993. ed. by Les Piegl.
[Gol95] Ronald N. Goldman. Identities for the B .il,. basis functions. volume V, chapter IV.2,
pages 163-167. Academic Press, Inc., 1995.
[NPL98] D. Nairn, J. Peters, and D. Lutterkort. -1 .." quantitative bounds on the distance
between a polynomial piece and its b6zier control polygon. Computer Aided Geometric
Design, 1998. to appear.

DAVID LUTTERKORT, DEPT. OF COMPUTER SCIENCE, PURDUE UNIVERSITY, WEST LAFAYETTE,
IN 47907-1398
E-mail address: lutter@cs.purdue.edu
URL: http://ww.cs.purdue.edu/people/lutterdc
JORG PETERS, DEPARTMENT OF COMPUTER AND INFORMATION SCIENCE, CSE ROOM 326,
UNIVERSITY OF FLORIDA, GAINESVILLE, FL 32611-6120
E-mail address: jorg0cise.ufl.edu
URL: http://www.cise.ufl.edu/j org


1This is a corrected version of a formula given in [Gol95]. The original formula is missing the
factor of 1/d! on the right hand side.




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