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 BSPLINES
A piecewise polynomial p of degree d in BSpline 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 wellknown recursion
formulas for the BSpline basis, see e.g. [dB93].
For .. '! ...... BSplines, 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 kth 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 = bk1 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 B6zierform 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 TR98013, 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,
Sai = t aki = +1 +1( ) t0* = o0
i i i
because of the linear precision of BSplines, tN = t.
These two qualities together prove that the functions 3ki given by
i
/Okit) = E (i j)aki
are all nonnegative: since Zi (ki = i iaki = 0 we can express ki3 as
k (i )M3 i
(j i) N i > k + 1.
The nonii. 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 nonnegative
Again, only finitely 11i i of the 3ki are nonzero. 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 BSPLINE 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 + d2)
This shows that q is a nonnegative 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 BSplines 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 BSplines, 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 = 2d 2 (2 d d bij = 2d d l)b
J=0 21 + 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+ bij.
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 2bij
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)= VbkNd2(t k) = bkNdd2(2t k)
k k
= Abk1Nd2(t k) = E Abk1Nd2(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 +ai2 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
[d2]
Vb2i+2 = 2 d (d ) vbi+l
[d/2] 1 d
jo
Ab2i+1 = 2 2 1) Ab
j=0
and for the odd indices
Vb2i+1 = 2d
j=o
[d/2] 
Ab2i = 2d
j=o
(d +l)Vbij
(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(d21)
^ J
2d1 and the
S(j = 2d2
2j I
2.1. Special cases. For quadratic BSplines, uniform refinement is ('I! ii,!:, al
gorithm and
b2i = 22(, t + bi)
b2i+1 = 22(bi, + ; 1.
This yields
1
A2b2i = A2b2i1 = A2bi_
4
Similarly, for cubic BSplines we have
62i = 23(bi2 +' Lt + bi)
b2i+1 = 23(4bi1 + 4bi).
and
1 1
A2 A2i= A2bi_ and A2b2i+1 = (A2bi_ + A2b).
4 8
UNIFORM BSPLINE BOUND
APPENDIX A. REPRESENTING t2 AS A UNIFORM BSPLINE
In general, monomials can be represented as BSplines through1
A
E tk+l () Nh
(oEnd i=1
Here, IId is the set of all permutations of the numbers {1,..., d}.
For uniform BSplines 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 2749. 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 163167. 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 479071398
Email 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 326116120
Email 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.
