Sains Malaysiana 39(4)(2010): 655–659

Pembinaan Permukaan Tertutup Licin oleh Fungsi Ball pada Kubus

(Construction of Smooth Closed Surfaces by Ball Functions on a Cube)

Diana Sirmayunie Mohd Nasir

Bahagian Hal Ehwal Akademik

Universiti Teknologi Mara Cawangan Perlis, 02600 Arau Perlis, Malaysia

Abd Rahni Mt Piah*

Pusat Pengajian Sains Matematik, Universiti Sains Malaysia

11800 USM Pulau Pinang, Malaysia

Received: 25 August 2009 / Accepted: 12 November 2009

ABSTRAK

Dalam Reka Bentuk Geometri Dibantu Komputer (CAGD), pembinaan permukaan secara asasnya dibentuk daripada koleksi tampalan permukaan, dengan syarat keselanjaran tertentu diletakkan di antara tampalan bersebelahan. Sungguhpun pada masa ini tampalan BŽzier hasil darab tensor digunakan secara meluas dalam kebanyakan sistem CAGD untuk memodel permukaan bentuk bebas, kaedah ini hanya boleh digunakan untuk memodel permukaan tertutup bergenus satu, iaitu permukaan yang setara dengan suatu torus. Permukaan dengan keselanjaran satah tangen dikenali sebagai permukaan licin secara geometri peringkat satu atau permukaan G1. Makalah ini mengemukakan satu kaedah pembinaan permukaan G1 yang mudah, iaitu permukaan bergenus sifar, dengan menakrif fungsi bikubik Ball pada muka kubus. Fungsi asas yang dibina mempunyai sokongan yang kecil dan hasil tambah fungsi asas adalah satu. Fungsi ini berguna untuk mereka bentuk, mencari hampiran dan menginterpolasi permukaan tertutup mudah yang genusnya adalah sifar. Kaedah pembinaan sebegini mula diperkenal oleh Goodman pada tahun 1991 yang telah menakrif fungsi splin-B teritlak bikuadratik pada mesy sisi empat mudah. Beberapa contoh permukaan/objek yang dihasilkan melalui kaedah pembinaan yang dikemukakan dipapar dalam makalah ini.

Kata kunci: Fungsi Ball; interpolasi; keselanjaran geometri; penghampiran; permukaan tertutup

ABSTRACT

In Computer Aided Geometric Design (CAGD), surface constructions are basically formed from collections of surface patches, by placing a certain continuity condition between adjacent patches. Even though tensor product BŽzier patches are currently used extensively in most CAGD systems to model free-form surfaces, this method can only be used to generate closed surface of genus one, i.e. a surface which is equivalent to a torus. A surface with tangent plane continuity is known as a first order geometrically smooth surface or a G1 surface. This paper presents a simple G1 surface construction method, i.e. a surface of genus zero, by defining Ball bicubic functions on faces of a cube. The constructed basis functions have small support and sum to one. The functions are useful for designing, approximating and interpolating a simple closed surface of genus zero. This construction method was first introduced by Goodman in 1991 who defined biquadratic generalised B-spline functions on faces of a simple quadrilateral mesh. Several examples of surfaces/objects which are constructed by the proposed method are presented in this paper.

Keywords: Approximation; Ball function; closed surface; geometric continuity; interpolation

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*Corresponding author; email: arahni@cs.usm.my