My research topic explores geometric theorems and algorithms focused on signed distance functions (SDFs) and distance-based techniques in computer graphics. My work introduces theoretical contributions like operations for distance-based surface modeling, including offset operations and constructive solid geometry (CSG) with signed distance function estimates. It also develops advanced ray tracing methods, such as enhanced sphere tracing and maximal disjoint cone searching for soft shadows, improving visualization with SDFs. Additionally, I focus on efficient geometric algorithms, including fast SDF rasterization for text rendering and collision detection using k-sided polytopes, offering both theoretical insights and practical applications for real-time graphics.