Study of Subdivision Curve Schemes and Their Impact on Computer-Aided Geometric Design and Computer Graphics

Abstract

Computer-Aided Geometric Design (CAGD) is a branch of applied mathematics that focuses on the computational modeling and representation of geometric shapes. It plays a crucial role in diverse fields such as geographic information systems, computer gaming, medical imaging, robotics, engineering, and traditional industries like automobile, aircraft, and ship design. A core challenge in CAGD is the creation of smooth curves and surfaces through efficient mathematical techniques. In recent years, subdivision schemes have emerged as a practical and elegant method for generating smooth limit curves. These techniques are widely used in computer animation (CA), CAGD, and computer graphics (CG) due to their simplicity, flexibility, and effectiveness. This study presents three prominent subdivision schemes which as the Chaikin Subdivision (CS), Corner-Cutting Subdivision (CSS), and Four-Point Subdivision (FPS) schemes. Each operates by refining an initial control polygon through iterative rules that add new points as weighted combinations of existing ones. Repeated application of these rules produces a limit curve with increasing smoothness. We begin by constructing various initial shapes—such as a jar, a mango, pi, and a car—and applying the CS scheme at multiple subdivision levels. Next, we apply the CSS scheme to shapes like a rabbit, a five-fingered hand, and a pi. Finally, the FPS scheme is applied to “U”, mug, and pi shapes. The results convincingly demonstrate the capability of these schemes to produce smooth and visually appealing curves with high precision. Looking ahead, our future research aims to develop and investigate non-uniform variants of these schemes. This direction seeks to improve adaptability for handling complex and irregular geometries, while maintaining the core properties of convergence, smoothness, and visual fidelity—ultimately expanding their utility in advanced geometric modeling.

J. Bangladesh Math. Soc. 45.1 (2025) 50–63