TY - JOUR

T1 - Topologically reliable approximation of composite Bézier curves

AU - Cho, Wonjoon

AU - Maekawa, Takashi

AU - Patrikalakis, Nicholas M.

N1 - Funding Information:
The authors appreciate the input by Prof. J. Peraire on the application of topologically reliable approximation for finite element meshing, the assistance of Dr. N. Sapidis with the literature identification in the early stages of this work and his comments and also the reviewers’ suggestions. Funding for this work was obtained in part from ONR and NSF under grant numbers NO001 4-94-1-1001 and DMI-9500394. Funding for the development of the nonlinear system solver used in this paper was obtained in part from MIT Sea Grant, ONR and NSF under grant numbers NA90AA-D-SG-424; NOOO14-91-J-1014 and N00014-94-1-1001: and DMI-9215411.

PY - 1996/8

Y1 - 1996/8

N2 - We present an efficient method of approximating a set of mutually nonintersecting simple composite planar and space Bézier curves within a prescribed tolerance using piecewise linear segments and ensuring the existence of a homeomorphism between the piecewise linear approximating seg-ments and the actual nonlinear curves. Equations and a robust solution method relying on the interval projected polyhedron algorithm to determine significant points of planar and space curves are described. Preliminary approximation is obtained by computing those significant points on the input curves. This preliminary approximation, providing the most significant geometric information of input curves, is especially valuable when a coarse approximation of good quality is required such as in finite element meshing applications. The main approximation, which ensures that the approximation error is within a user specified tolerance, is next performed using adaptive subdivision. A convex hull method is effectively employed to compute the approximation error. We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. For each pair of linear approximating segments, an intersection check is performed to identify possible inappropriate intersections. If these inappropriate intersections exist, further local refinement of the approximation is performed. A bucketing technique is used to identify the inappropriate intersections, which runs in O(n) time on the average where n is the number of linear approximating segments. Our approximation scheme is also applied to interval composite Bézier curves.

AB - We present an efficient method of approximating a set of mutually nonintersecting simple composite planar and space Bézier curves within a prescribed tolerance using piecewise linear segments and ensuring the existence of a homeomorphism between the piecewise linear approximating seg-ments and the actual nonlinear curves. Equations and a robust solution method relying on the interval projected polyhedron algorithm to determine significant points of planar and space curves are described. Preliminary approximation is obtained by computing those significant points on the input curves. This preliminary approximation, providing the most significant geometric information of input curves, is especially valuable when a coarse approximation of good quality is required such as in finite element meshing applications. The main approximation, which ensures that the approximation error is within a user specified tolerance, is next performed using adaptive subdivision. A convex hull method is effectively employed to compute the approximation error. We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. For each pair of linear approximating segments, an intersection check is performed to identify possible inappropriate intersections. If these inappropriate intersections exist, further local refinement of the approximation is performed. A bucketing technique is used to identify the inappropriate intersections, which runs in O(n) time on the average where n is the number of linear approximating segments. Our approximation scheme is also applied to interval composite Bézier curves.

KW - Bucketing technique

KW - Homeomorphism

KW - Interval Bézier curves

KW - Piecewise linear approximation

KW - Significant points

KW - Subdivision method

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U2 - 10.1016/0167-8396(95)00042-9

DO - 10.1016/0167-8396(95)00042-9

M3 - Article

AN - SCOPUS:0030215146

VL - 13

SP - 497

EP - 520

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 6

ER -