Computergeometrie

Computergeometrie, commonly known in English as computational geometry, is a field of computer science and mathematics that studies algorithms and data structures for solving geometric problems. It focuses on the design, analysis and implementation of procedures that handle points, lines, polygons, polyhedra and higher-dimensional geometric objects, emphasizing correctness, efficiency and robustness.

The discipline emerged in the 1970s from applications in computer graphics, robotics and geographic information systems, and matured through foundational work on convex hulls, Voronoi diagrams and Delaunay triangulations. Typical problems include nearest neighbor search, range searching, motion planning, intersection detection, polygon triangulation and mesh generation.

Algorithmic approaches in Computergeometrie range from exact symbolic methods to numeric and randomized techniques. Complexity considerations are central: worst-case, average-case and output-sensitive bounds guide algorithm selection and refinement. Practical implementations must also address numerical precision, degeneracies and data structure efficiency.

Applications span computer graphics and visualization, geographic information systems, robotics and path planning, computer-aided design and manufacturing, scientific computing and spatial databases. Libraries and software packages — both academic and commercial — provide routines for common tasks such as convex hull computation, spatial indexing and mesh processing.

Computergeometrie intersects with computational topology, numerical analysis and optimization. Ongoing research explores scalable algorithms for high-dimensional data, robust computation under limited precision and integration with machine learning for geometry-aware applications.