Positioning non-regular polygons over 3D surfaces

Abstract

Positioning non-regular polygons on 3D surfaces while adhering to complex constraints is a key challenge in computational geometry and surface modeling. This thesis presents three algorithms designed to optimize the arrangement of non-regular polygons on a given 3D surface, represented as a matrix of z-values (elevation) at discrete (x, y) coordinates. The algorithms address multiple constraints, including polygon orientation, non-overlapping placement, and surface topology. Additionally, statistical constraints on the mean height and standard deviation of the z-values within the entire area of each polygon ensure that the placements conform to specific surface features. The 3D surface is divided into quadrants, each with a preferred polygon orientation. OpenCV is utilized for both collision detection and masking, ensuring that polygons do not overlap and are placed within surface boundaries. The algorithms manage polygon projection onto the surface, handle overlaps, and apply orientation constraints through ro- tation matrices. Further, user-defined thresholds for mean height and standard deviation guide the selection of valid placements, ensuring alignment with the surface's geometric properties. The algorithms were evaluated on different surfaces with various topological features, demonstrating their effectiveness in maximizing polygon coverage while satisfying all con- straints. This work has significant applications in surface modeling, geographic informa- tion systems, and industrial design, where efficient and constrained polygon placement is essential.