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

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