Modeling in Computer Graphics through Visualizable Mathematical Structures - Applications in Τeaching Mathematics

Abstract

In a world dominated by digital information and visual representation, this doctoral dissertation explores the deep and bidirectional relationship between computer graphics and mathematical structures, with a focus on how these structures can be visualized and employed as tools for teaching Mathematics. Since the dawn of computer science, graphics have represented a cutting-edge field—enabling the creation of virtual worlds, the simulation of complex phenomena, and novel ways to interact with data. Behind every realistic image, every three-dimensional model, and every animation lie powerful mathematical tools. The beauty and usefulness of these applications stem from their capacity to transform abstract mathematical concepts into tangible, visual representations. This dissertation does not limit itself to identifying and describing the mathematical principles underlying computer graphics. Rather, it introduces a systematic framework of evaluation criteria to precisely determine which mathematical models—from algebraic and differential manifolds to topological structures—are feasible, computationally viable, and cognitively meaningful to visualize. In this way, the dissertation goes beyond a descriptive catalog of principles and offers an instrumental foundation that enables researchers to determine, clearly and systematically, which mathematical structures are both valuable and possible to transition from theoretical conception to pictorial representation. Over the past two decades, the use of computer graphics has expanded across many domains. The rapid evolution of computer graphics and informatics has inevitably impacted the sensitive field of education. Integrating computer graphics into Mathematics instruction presents a unique opportunity to overcome longstanding challenges in understanding abstract concepts. Through interactive visualizations, students can “see” and “interact” with mathematical structures, experiment with parameters, and uncover relationships that would otherwise remain hidden. This doctoral dissertation successfully bridges the gap between theoretical mathematical knowledge and practical applications in computer graphics by proposing and developing visualizable mathematical structures that serve as teaching tools. It goes beyond theoretical discussion by presenting specific educational scenarios that utilize these structures to enhance the comprehension of challenging mathematical concepts. Selected modeling examples are analyzed, detailing their mathematical foundations and illustrating how they can be translated into interactive, educational applications. In summary, this dissertation addresses: (a) the definition of control criteria for visualizable mathematical structures, and (b) the interdisciplinary development of a new framework for understanding and teaching Mathematics through the power of computer graphics—aiming to transform the way students perceive and engage with abstract mathematical ideas.