Sains Malaysiana 42(7)(2013): 989–997

Lengkung Peralihan Licin dengan Kuartik Bezier Satahan

(Smooth Transition Curve by Planar Bezier Quartic)

Azhar Ahmad

Fakulti Sains Dan Matematik, Universiti Pendidikan Sultan Idris, 35900 Tanjung Malim,

Perak, Malaysia

Jamaludin Md Ali

Sekolah Sains Matematik, Universiti Sains Malaysia, 11800 Minden, Pulau Pinang

Malaysia

Received: 21 July 2011/Accepted: 10 February 2013

ABSTRAK

Kertas ini membincangkan satu keluarga lengkung peralihan satahan licin yang dibina menggunakan dua lingkaran kuartik Bezier. Lingkaran kuartik yang diperkenalkan ini mempunyai enam darjah kebebasan yang memberi kelebihan untuk mempelbagaikan rupa bentuk lingkaran di dalam selangnya. Kelicinan adalah dirujuk melalui ciri-ciri keselanjaran geometri G₂ dan kelengkungan monoton yang dimiliki oleh sesuatu lengkung peralihan. Oleh sebab lengkungan ini tidak memiliki sebarang juring, gelung dan titik lengkok balas, di samping tanpa perubahan kelengkungan yang mendadak maka ia amat sesuai untuk aplikasi tertentu dalam Reka Bentuk Berbantukan Komputer (RBK) bagi memenuhi keperluan estatis serta kepentingan fungsinya seperti reka bentuk produk industri, trajektori robot tidak holonomi dan juga reka bentuk mendatar landasan kereta api serta lebuh raya. Sebagai suatu perwakilan polinomial, lingkaran kuartik Bezier ini mampu digabungjalinkan ke dalam sistem RBK yang kebanyakannya berasaskan perwakilan NURBS (nonuniform rational B-splines).

Kata kunci: Kelengkungan monoton; lengkung peralihan; lingkaran kuartik bezier

ABSTRACT

This paper reports on a family of smooth transition curve rendered by a pair of Bezier quartic spiral. The family of Bezier quartic spiral form which was introduced has six degrees of freedom and will give advantage on diversity of the transition curve in the given constraints. Fair curves often referred on geometrical continuity G₂ and monotonically of curvature of the transition curves. Since these curves have no cusps, loops and the inflection points, and free from abrupt chance of curvature, therefore it is suitable for some particular application in Computer Aided Design (CAD) in terms of aesthetical and functional values, especially in the industrial products design, trajectory of non-holonomic robot, as well as the horizontal route designs of railways and highways. Since Bezier quartic spiral is polynomial, it can be conveniently incorporated in CAD systems in the form of NURBS (nonuniform rational B-splines) representation.

Keywords: Bezier quartic spiral; monotone curvature; transition curve

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*Corresponding author; email: azhar.ahmad@fsmt.upsi.edu.my