Are two curves the same?

Diana Pekerman, Joon Kyung Seong, Gershon Elber, Myung Soo Kim

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We present an algorithm for answering the following fundamental question: Given two arbitrary (piecewise) polynomial curves, are they the same? This basic CAGD question is answered by first reducing the two curves into canonical irreducible forms. This is done by reversing the processes of knot refinement, degree raising, and composition. The two curves are then compared in their irreducible forms and their shared domains, if any, are identified. The ability to answer this fundamental identity question will be a boon for numerous applications. In this paper, we demonstrate a few such applications. The algorithm allows one to identify two boundary curves (shared as a common seam) between two different surfaces as an identical curve (or not) even when they are represented differently. Moreover, we show that reparameterization is insecure as a watermarking method, which invalidates the proposal of [8].

Original languageEnglish
Pages (from-to)85-94
Number of pages10
JournalComputer-Aided Design and Applications
Issue number1-4
Publication statusPublished - 2005
Externally publishedYes

Bibliographical note

Funding Information:
All the algorithms and figures presented in this paper were implemented and created using the IRIT solid modeling system [4] developed at the Technion, Israel. This work was supported in part by the Israeli Ministry of Science Grant No. 01–01–01509, in part by the Israel Science Foundation (grant No. 857/04), and in part by the European FP6 NoE grant 506766 (AIM@SHAPE), in part by the Korean Ministry of Information and Communication (MIC) under the Program of IT Research Center on CGVR, in part by the Korean Ministry of Science and Technology (MOST) under the Korean-Israeli binational research grant, and in part by grant No. R01-2002-000-00512-0 from the Basic Research Program of the Korea Science and Engineering Foundation (KOSEF).


  • Composition
  • Curve matching
  • Decomposition
  • Degree-raising
  • Degree-reduction
  • Knot refinement
  • Knot removal
  • Polynomials
  • Rationals
  • Water-marking

ASJC Scopus subject areas

  • Computational Mechanics
  • Computer Graphics and Computer-Aided Design
  • Computational Mathematics


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