APPLYING ANALYTICAL GEOMETRY IN COMPUTER GRAPHICS: A DEEP DIVE WITH VISUALIZATIONS (ILLUSTRATIVE REPRESENTATION)

Abstract

This article explores the fundamental role of analytical geometry in computer graphics, providing a deep dive into its core principles and applications with illustrative visualizations. The paper begins by establishing foundational concepts of coordinate systems (Cartesian, cylindrical, spherical) and vector operations, essential for defining object positions, transformations (translation, rotation, scaling), and calculating distances and angles. It then delves into the equations defining various geometric objects, from lines and circles/ellipses to planes and higher-order surfaces, illustrating how these equations underpin the creation and manipulation of graphical elements. The article further examines object transformations using transformation matrices and projections (central and parallel), crucial for rendering 3D objects onto a 2D screen. Finally, it touches upon advanced topics such as splines, Bézier curves, and collision detection, highlighting the continued importance of analytical geometry in generating complex shapes and simulating interactions within computer-generated environments. Throughout, visual representations enhance understanding of the key concepts and their applications.