Computer Aided Geometric Design 18 (2001) 73–76www.elsevier.com/locate/comaid

Erratum

Erratum to: “A closed algebraic interpolation curve”,Computer Aided Geometric Design 17 (7) (2000) 631–642✩

Wolfgang SchusterUniversitat Tübingen, Deutsches Institut für Fernstudienforschung, Konrad Adenauerstrasse 40,

72072 Tübingen, Germany

Pages 631 and 632 of the original version are erroneous due to a technical problem. Thecorrect version is given below.

Abstract: We introduce a new interpolation method employing a closed algebraic curve.Keywords: Interpolation; Fourier series; Chebyshev polynomials; Roller coaster

Introduction: This note describes a trigonometric interpolation method to connect agiven point sequence inRd by an algebraic and therefore smooth closed curve. Theinterpolation formula is of Lagrangian type and seems to be new. In contrast to splineinterpolation our method does not work piecewise but global. Nevertheless the curveconnects the given points in a quite natural manner preserving the given order and withoutovershooting tendency. The curves are, like spline curves, stable in the following sense:changing the position of a single point of the given sequence only changes the curve in theneighbourhoodof this point essentially. Besides its trigonometric parametrization the curvepossesses a numerically efficient parametrization on the basis of Chebyshev polynomialsof the second kind. To what extend the new interpolation method can replace the splinemethod, the experts may decide.

1. Finite Fourier series approach A smooth closed curveX(t) is to be drawn runningthroughn = 2m + 1, m � 0, ordered points ofRd , that is through the vertices of apolygonX ⊂ Rd with an odd number of vertices. The position vectors of the vertices areX0,X1, . . . ,Xn−1. We requireX(

jn) = Xj , j = 0,1, . . . , n − 1. It is natural to try a finite

Fourier series approach

✩ PII of original article: S0167-8396(00)00019-4.E-mail address: schuster@diff.uni-tuebingen.de (W. Schuster).

0167-8396/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0167-8396(00)00037-6

74 W. Schuster / Computer Aided Geometric Design 18 (2001) 73–76

X(t) =m∑

k=−m

Ake2π ikt (1)

with n coefficient vectorsAk ∈ Cd to be determined.SinceX(

jn) = Xj , the following n conditions have to be fulfilled by the coefficient

vectors:m∑

k=−m

Akλjk = Xj , λ = e

2π in , j = 0,1, . . . , n − 1. (2)

We consider then coefficient vectorsA−m, . . . ,Am as unknowns of this system ofn linearequations. After multiplying thej th row of (2) with 1

nλ−j l , −m � l � m (l fixed) and

summing up the rows one obtains the solutions

Al = 1

n

n−1∑

j=0

λ−j lXj . (3)

Substituting the coefficient vectorsAk in the Fourier series (1) by (3) gives the parameterrepresentation

X(t) =n−1∑

j=0

1

n

m∑

k=−m

λ−kj e2π iktXj (4)

of the curve wanted. Apparently the conditionX( ln) = Xl is fulfilled. The functions

σn,j (t) = 1

n

m∑

k=−m

λ−kj e2π ikt = 1

n

m∑

k=−m

e2π ik(t− jn ), j = 0,1, . . . , n − 1, (5)

are real and can be written in the form

σn,j (t) = sinπn(t − jn)

nsinπ(t − jn), (6)

using the summation formula for geometric series. In retrospect one recognizes that theassumption, the number of points to be odd, guarantees the functionsσn,j to be real.

We summarize our result:

Theorem 1. Let X0,X1, . . . ,Xn−1 be an odd number of points in Rd . Then

X(t) =n−1∑

j=0

σn,j (t)Xj , 0 � t � 1, (7)

is the parameter representation of a smooth closed curve going through the points Xj and

X(jn) = Xj , j = 0,1, . . . , n − 1, holds.

The curve with parameter representation (7) may be called aσ -curve. As ourargumentation has shown, the finite Fourier series (1) is uniquely determined by theconditionX(j/n) = Xj .

W. Schuster / Computer Aided Geometric Design 18 (2001) 73–76 75

A clear picture of the Figs. 1, 2, 4, 5, and 6 are herewith given by reprint.

Fig. 1.σ -curves throughn = 5, 15, 25, 35 points.

Fig. 2. The functionσn,0(t) for n = 5, 15, 25, 35 points.

76 W. Schuster / Computer Aided Geometric Design 18 (2001) 73–76

Fig. 4. Changing a single vertex of the polygon.

Fig. 5. Part of aσ -curve going through an even number of given points with three control points.

Fig. 6. The tangent of the curve at a vertex of the polygon.