Sains Malaysiana
37(4): 421-427 (2008)
Approximating Offset Curves by Rational Bézier
Cubics and Quartics
(Penganggaran Lengkung Ofset dengan Kubik Nisbah Bézier
Dan Kuartik Nisbah Bézier)
Chong Nyuk Sian
Department of Mathematics, Faculty of Science and Technology
Universiti Malaysia Terengganu, 21030 Kuala Terengganu
Terengganu, Malaysia
Received: 25 January 2008 / Accepted: 24 March 2008
ABSTRACT
Offset curves arise in a variety of industrial applications such
as robot’s path planning and numerical control machining
in the textile, shoe and automobile industries. Rational
curves, in particular the rational cubics, are widely accepted
as a standard representation for design problems and geometric
modellers but their offset curves are in general not rational.
Given a rational cubic or quartic spline, we present two
local methods to approximate its offset curve using a rational
Bézier spline of the same degree. This approximate offset
curve interpolates the positions and unit tangents at both
ends of the exact offset curve segments and its curvatures
at these endpoints are consistent with the offset distance
and the corresponding curvatures of the given curve. It
has second order geometric continuity if the given curve
is so. The accuracy of the approximation can be refined
by a local iterative subdivision process.
Keywords: Approximation; offset curve; rational Bézier curve
ABSTRAK
Lengkung
ofset timbul dalam pelbagai jenis aplikasi industri seperti
perancangan laluan robot dan mesin kawalan berangka dalam
industri tekstil, kasut dan automobil. Lengkung nisbah,
khususnya lengkung kubik nisbah telah diterima secara meluas
sebagai suatu perwakilan piawai bagi masalah-masalah reka
bentuk dan pemodelan geometri tetapi secara umumnya, lengkung
ofsetnya adalah bukan nisbah. Diberi suatu lengkung kubik
atau kuartik nisbah, kita mewakilkan dua kaedah setempat
untuk menganggar lengkung ofsetnya dengan menggunakan suatu
splin Bézier nisbah yang sama darjah. Lengkung hampiran
ofset ini menginterpolasi kedudukan dan tangen unit di kedua-dua
titik hujung tembereng ofset sebenar dan kelengkungannya
pada titik-titik hujung ini adalah konsisten dengan jarak
ofset serta kelengkungan yang sepadan dengan lengkung yang
diberi. Ia mempunyai keselanjaran geometri berdarjah dua
jika lengkung yang diberi juga bersifat sedemikian. Kejituan
penganggaran boleh diperhaluskan melalui proses lelaran
sub-bahagian setempat.
Kata
kunci: Lengkung ofset; lengkung nisbah Bézier; penganggaran
REFERENCES/RUJUKAN
Farin, G. 1996. Curves and Surfaces for Computer
Aided Geomteric Design, 4th ed.
Academic Press, Inc, Boston.
Farouki, R.T. and Sakkalis, T. 1990. Pythagorean hodographs.
IBM J. Research & Development 34: 736-752.
Hoschek, J. 1988. Spline approximation of offset curves.
Computer Aided Geometric Design 5: 33-40.
Klass, R. 1983. An offset spline approximation for plane
cubic splines. Computer Aided Design 15: 297-299.
Lee, In-Kwon, Kim, Myung-Soo and Elber, Gershon. 1996.
Planar curve offset based on circle approximation. Computer
Aided Design 28: 617-630.
Pham, B. 1988. Offset approximation of uniform B-splines.
Computer Aided Design 20:
471-474.
Pham, B. 1992. Offset curves and surfaces: a brief survey.
Computer Aided Design 24:
223-229.
Piegl, L. 1987. Interactive data interpolation by rational
Bézier curves. IEEE Computer Graphics & Applications7:
45-58.
Tiller W. and Hanson, E.G. 1984. Offsets of two-dimensional
profiles. IEEE Computer Graphics & Applications
4: 36-46.