Are two curves the same?

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

doi:10.1080/16864360.2005.10738356

Are two curves the same?

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

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

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 language	English
Pages (from-to)	85-94
Number of pages	10
Journal	Computer-Aided Design and Applications
Volume	2
Issue number	1-4
DOIs	https://doi.org/10.1080/16864360.2005.10738356
Publication status	Published - 2005
Externally published	Yes

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).

Keywords

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

Access to Document

10.1080/16864360.2005.10738356

Cite this

@article{79a6f80906404bcf803084761959300d,

title = "Are two curves the same?",

abstract = "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].",

keywords = "Composition, Curve matching, Decomposition, Degree-raising, Degree-reduction, Knot refinement, Knot removal, Polynomials, Rationals, Water-marking",

author = "Diana Pekerman and Seong, {Joon Kyung} and Gershon Elber and Kim, {Myung Soo}",

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).",

year = "2005",

doi = "10.1080/16864360.2005.10738356",

language = "English",

volume = "2",

pages = "85--94",

journal = "Computer-Aided Design and Applications",

issn = "1686-4360",