Approximation of Freeform Surfaces with Polyhedra Composed of Congruent Triangles

Abstract

The article presents the structure and stages of implementing a computer geometric modeling program designed to solve a discrete geometry problem: the approximation of free-form surfaces using polyhedra with groups of congruent faces. Solving this problem reduces the costs of physical construction of the specified surface in the form of building facades, particularly in the new architectural trend known as "parametricism." The optimization method utilized in the program enables the creation of geometric models that are valuable in architecture, industrial design, and computer graphics. A key component of the program is a genetic algorithm, one of the evolutionary computation methods. The paper provides a detailed analysis of the genetic algorithm's parameters, including the number of generations, population size, mutation and crossover probabilities. The optimal set of parameters for the program is determined experimentally. Experimental results demonstrate the effectiveness of various algorithm configurations. The findings reveal that the optimization method used minimizes the number of distinct congruent polyhedral faces. The advantages of the method are evaluated within the program’s constraints, as well as its limitations, such as computational complexity and the need for parameter tuning. The conclusion discusses prospects for future research, including enhancing the algorithm's efficiency, developing new approaches for normalizing and discretizing input data, controlling and modifying the topology of the resulting polyhedra, and exploring alternative optimization methods.