QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXI, NUMBER 3
SEPTEMBER 2003, PAGES 475-484
MORE-OR-LESS-UNIFORM SAMPLING
AND LENGTHS OF CURVES
By
LYLE NOAKES (Department of Mathematics and Statistics, The University of Western Australia,
35 Stirling Highway, Crawley WA 6009, Australia)
RYSZARD KOZERA (Department of Computer Science and Software Engineering, The University
of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia)
Abstract. More-or-less-uniform samples are introduced and used to estimate lengths
of smooth regular strictly convex curves in R2. Quartic convergence is proved and illus-
trated by examples.
1. Introduction. The problem of measuring the length of a curve has a long history
in mathematics, dating back to ancient geometry. In particular, Archimedes and Liu
Hui [11] estimated the length of a circular curve. Jordan, Peano and others introduced
digitizations of sets in R2 and R3 for the purpose of different feature measurements such
as perimeters (see, e.g., [4]). Related historical and contemporary work can be found in
[2], [3], [6], [7], [9], [10], and [12],
Let 7: [0,T] —> R™ be a smooth regular curve; namely, 7 is Ck for some k > 1 and
7(t) 7^ 0 for all t £ [0, T\. The length of 7 is defined to be
dh) = f \\i(t)\\dt,
Jo
where 7 is the derivative of 7, and || • || is the Euclidean norm. Consider the problem of
estimating d(7) from an ordered (m + l)-tuple
Q = (qo,qi,---,qm)
of points in Rn, where qi = j{ti), and 0 = to < ti <■■■< ti <■•■< tm = T. Depending
on what is known about the ti, the problem may be straightforward or unsolvable.
Example 1. Let 7 be Cr+2, where r is a positive integer, and take m to be a multiple
of r. Then Q gives y (r + l)-tuples of the form
(*70) 91; • • • > 9r)> (<?i-)9r+l> • • • j </2r)) • • • j (? m—ri Qm—r+1? • • • ■> Qm)>
Received April 10, 2001.
2000 Mathematics Subject Classification. Primary 65Dxx.
E-mail address: lyleOmaths. uwa. edu. au
E-mail address: ryszardOcs.uwa.edu.au
475
©2003 Brown University
476 LYLE NOAKES and RYSZARD KOZERA
The jth (r + l)-tuple can be interpolated by a polynomial 7j: [t(j-i)r,tjr] —* R™ of
degree r, and the track-sum 7 of the 7j is everywhere continuous and C°° except at the
knot points tr,t2r> ■ ■ ■ ,tm-r■ Suppose that sampling is uniform: ti = ^ for 0 < i < m.
The errors in Lagrange interpolation are best studied using Lemma 2.1 in Sec. 2 of Part
I of [5]. We find that 7(t) = 7(t) + 0(m}+1) for t € [0, T], and 7 = 7(t) + O(^) for
t 7^ tr,t2r, ■ ■ ■; im—r• Consequently, d(7) — d(7) = 0{~). This error can be shown to
be 0(m}+ 2) or 0( *+1) according as r is even or odd [8].
Example 2. Let t\ = ^ and = ti + ^ for 2 < z < m. Q gives only endpoint
information for 7 over [0, j], and therefore does not even determine an upper bound on
d(j) as m —> 00.
An intermediate situation is where the tt are not given, but sampled more-or-less
uniformly in the following sense.
Definition 1. Sampling is more-or-less uniform when there are constants 0 < Ki <
Ku such that, for any sufficiently large integer m, and any 1 < i < m,
Ki Ku
— <U-U-i < —■
m m
The uniform sampling of Example 1 is more-or-less uniform, and the sampling in
Example 2 is not. With more-or-less uniform sampling, increments between successive
parameters are neither large nor small in proportion to ~i. Then, just as for the uniform
sampling of Example 1, piecewise-linear interpolation between sample points approxi-
mates the image of 7 to O(^), and d(7) to 0(^2 )■ However, use of piecewise-quadratic
instead of piecewise-linear can lead to unfortunate results, because of the need to es-
timate1 the parameters ti for 0 < i < m. If we guess U = then the resulting
piecewise-quadratic 7: [0,1] —> Rn is sometimes informative [7], [8], and sometimes not.
Example 3. For 0 < i < m, set U = ^3t+^n1^ ■ Then sampling is more-or-less
uniform, with Ki = j, Ku = Let 7: [0,7r] —* M2 be the parametrization 7(t) =
(cos t, sin t) of the unit semicircle in the upper half-plane. When m is small, the image of
7 does not much resemble a semicircle, as in Fig. 1 where m = 3 and d{^)—d{^) = 0.0601.
The error in length estimate with piecewise-linear interpolation is —0.0712. When m is
large the image of 7 looks semicircular, as in Fig. 2 where m = 30. In this case, however,
d(7) — d(7) = 0.1194, an error nearly twice as large as for m = 6. Even piecewise-linear
interpolation with 31 points gives a better estimate, with error —0.0033. Indeed, as
m increases (at least for m < 100), piecewise-quadratic interpolation tends to increase
errors of length estimates. Linear interpolation is better, but not impressive.
Example 4. For 0 < i < m, let 1,t be a random number (according to some distri-
bution) in the interval [^3l^T, Then sampling is more-or-less uniform, with
Ku, Ki as in Example 3.
Example 5. Choose 6 > 0 and 0 < Li < Lu. Set so = 0. For 1 < i < m, choose
Si G [^, independently from (say) the uniform distribution. Define Sj = Sj_ 1 + 6i
for i = 1,2,..., m. The expectation of sm is Lv+Li ancj the standard deviation ■
1In Example 1 these were assumed to be given.
MORE-OR-LESS-UNIFORM SAMPLING AND LENGTHS OF CURVES 477
Fig. 1. 7 data points, with 3 successive triples interpolated by
piecewise-quadratics, giving length estimate -n + 0.0601035 for the
semicircle (shown dashed).
Fig. 2. 31 data points, with 15 successive triples interpolated by
piecewise-quadratics, giving length estimate n + 0.119407 for the
semicircle.
So if m is large, sm ~ Lv+Li with high probability. For 0 < i < m. define ti = Set
^ sm
2 LtT 2LUT
l~Lu + Lt ' Ku~ Lu + U+e'
Then with high probability for m large, the sampling (to, ti, t2, ■ ■ ■, tm) from [0, T] is
more-or-less uniform with constants K[,KU.
More-or-less uniform sampling is invariant with respect to reparameterizations;
namely, if 0: [0, T\ —> [0, T] is an order-preserving C1 diffeomorphism, and if (to, t\,... ,tm)
are sampled more-or-less uniformly, then so are (cp(to),4>(ti),... ,<p(tm)). So reparame-
terizations lead to further examples from the ones already given. To state our main
result, first take n = 2 and suppose that 7 is C4 and (without loss) parameterized by
arc-length; namely, ||7|| is identically 1. The curvature of 7 is defined as
k(t) = det(M(t)) ,
where M(t) is the 2x2 matrix with columns ^(t),^(t). When k(t) ^ 0 for all t G [0, T],
7 is said to be strictly convex.
478 LYLE NOAKES and RYSZARD KOZERA
Theorem 1. Let 7: [0, T] —> 1R2 be strictly convex and suppose that sampling is more-
or-less uniform. Then, for some d(Q), calculable in terms of Q,
d(Q) = d(-y) + O
TO'
In Sections 2, 3 we prove Theorem 1, constructing d(Q) as a sum of lengths of qua-
dratic arcs interpolating quadruples of sample points. In Sec. 4, some examples are
given, showing that the quartic convergence of Theorem 1 is the best possible for our
construction.
Note added in proof. The authors have recently become aware of [13] which contains
work on closely related problems and other interesting references.
2. Quadratics interpolating quadruples. Let Q be sampled more-or-less uni-
formly from 7, and suppose (without loss) that to is a positive integer multiple of 3.
For each quadruple (qi, qi+i, qi+2, qi+3), where 0 < i < m — 3, define ao,ai,a2 £ M2 and
Ql(s) = a0 + a\s + a2S2 by
Ql(0) = qi , Ql{ 1) = qi+i , Ql(a) = ql+2 , and Ql(fi) = ql+3.
Then ao = qi, = qi+i — do — ai, and we obtain two vector equations:
aia + (pi - ai)a2 = pa and ax(5 + (pi - ai)/32 = p0 , (1)
where (pi,pa,pp) = (qi+1 - qi, qt+2 - qt, Qi+3 - qi)- Then (1) amounts to four quadratic
scalar equations in four scalar unknowns a\ = (an,a^),a,/?. Set
c= -det(pa,p0) , d = -det(p0,pi)/c , e = - det(pQ,pi)/c ,
where c,d,e ^ 0 by strict convexity, and define
Pi = \Je(l + d — e)/d , p2 = \/d(l + d - e)/e.
Then (1) has two solutions (as can be verified by substitution):
1 a \ (I+PI1I+P2) , a ^ (1 — Pi» 1 — P2)
(a+,/?+) = , (a_,/3_) = , (2)
6 — d 6 — d
provided pi, p2 are real and d — e / 0. We now justify these assumptions and show that
precisely one of (2) satisfies the additional condition
1 <a<(3. (3)
It suffices2 to deal with the case where k(t) < 0 for all t e [0, T]. Then it is rather
apparent, for geometrical reasons, that 1 + d — e, —d, —e, and e — d are all positive
asymptotically. Alternatively, these facts can be proved (and sharper estimates obtained)
by Mathematica calculations, as in Lemma 1 below. Define
l{t) =
It?)
k{t)
2The other case, where k(t) is everywhere positive, is dealt with by considering the reversed curve
7r(<) = (71 (T - 0.72(7' - t)).
MORE-OR-LESS-UNIFORM SAMPLING AND LENGTHS OF CURVES 479
Then, using Taylor's theorem, for t,u 6 3],
det(7(t)-ft,7(u)-gi) = fc-—^)(M ^ ^ + ^ _ 2t. +£)0 +O , (4)
where k, I are evaluated at tn.
Lemma 1.
, „ , ((ti+2 - U)(1 + i(t'V'+l)), (ti+3 - ti)(i + ;(ti+36"ti+l))) , ^ ( \\ ,c,
(a+'^+)~ +UWJ■ (5)
Proof. By (4),
C = -fc(ti+2 ~ *»)(*»+3 ~ *«)(*»+3 ~ *i+2) ^ + (t.+3 _ ^ + t.+a)^ + 0 (^_L ^ )
Cd = -fc(*<+3 ~ *i)(*i+l ~ ~ W3) ^ + (ti+i _ + t.+s)0 + G ^ ^
ce = _fc(^ -*0(^+1-^(^+1-^+2) ^ + (t.+i _ 2ti + t.+2)^ + 0 ^
Consequently,
('-"•«-^4)+°(^
The lemma follows from these two equations. The detailed calculation can be viewed at
the URL address http://www.cs.uwa.edu.au/~ryszard/4points/. □
We continue with the assumption that k(t) < 0 for all t. Then (3) follows from (5) for
to large with (a, (3) = (a+, /?+)• Then, for 0 < s < /?, Ql(s) = qi + a\s + 02s2, where
pa - a2pi Pa ~ (32Pi , api - pa (3px - pp
ai = 2 = ~h and "2 = 2 = 02"- (6)
Q! — Q! P — P Q — GI P — P
Lemma 2.
ai — (U+i — U)j{ti) ( 1 + O ( •—^ + O (—5- I ,
°2 - {ti+\ti) m f1+0 (£))+0 G^)i{ti)+0 (i
Proof. From (5),
/ 2 „ „2\ {ti+2 2 ti+2) ^ 1 ,
(a ,a - a ) = - —~z 1- (J — .
(ti+1 - U)2 m