Work

Parametric Surfaces for Elliptic and Hyperbolic Geometries

Year: 2025

Type: article

Abstract: Background/Objectives: In computer graphics, virtual worlds are constructed and visualized through algorithmic processes. These environments are typically populated with objects defined by mathematical models, traditionally based on Euclidean geometry. However, there is increasing interest in exploring non-Euclidean geometries, which require adaptations of the modeling techniques used in Euclidean spaces. Methods: This paper focuses on defining parametric curves and surfaces within elliptic and hyperbolic geometries. We explore free-form splines interpreted as hierarchical motions along geodesics. Translation, rotation, and ruling are managed through supplementary curves to generate surfaces. We also discuss how to compute normal vectors, which are essential for animation and lighting. The rendering approach we adopt aligns with physical principles, assuming that light follows geodesic paths. Results: We extend the Kochanek–Bartels spline to both elliptic and hyperbolic geometries using a sequence of geodesic-based interpolations. Simple recursive formulas are introduced for derivative calculations. With well-defined translation and rotation in these curved spaces, we demonstrate the creation of ruled, extruded, and rotational surfaces. These results are showcased through a virtual reality application designed to navigate and visualize non-Euclidean spaces.

Source: Mathematics

DOI: 10.3390/math13213403

Authors: László Szirmay-Kalos, András Fridvalszky, László Szécsi, Márton Vaitkus

Institution: Budapest University of Technology and Economics

Language: English

Topic: Computer Graphics and Visualization Techniques

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: National Research, Development and Innovation Office