Geometry & Spatial Modeling

Geometry and spatial modeling techniques deal with mathematical problems involving points, lines, polygons, and other geometric objects. These algorithms are crucial for computer graphics, computational geometry, and spatial data analysis.

Techniques in This Category

Technique	Description	Link
Points & Lines	Cross product, orientation, segment intersection.	Points & Lines
Distance Functions	Euclidean, Manhattan, point–line, point–segment.	Distance Functions
Polygon Area	Shoelace formula, signed area, Pick’s theorem.	Polygon Area
Complex Numbers	std::complex for rotation, angles, and plane geometry.	Complex Numbers
Closest Pair	Minimum distance between two points in a set; \(O(n \log n)\) approaches.	Closest Pair
Sweep Line	Process geometric events in sorted order (intersections, unions, etc.).	Sweep Line
Convex Hull	Minimal convex polygon containing all points.	Convex Hull

When to Use Geometric Algorithms

Geometric techniques are essential when: - Working with spatial data or coordinates - Solving problems involving points, lines, or polygons - Need to determine geometric relationships - Optimizing spatial queries or computations

Common Geometric Problems

• Point-in-Polygon: Determining if a point lies inside a polygon
• Line Intersection: Finding intersection points between lines or line segments
• Convex Hull: Finding the smallest convex polygon containing all points
• Closest Pair: Finding the two closest points in a set
• Area Calculations: Computing areas of polygons or regions
• Spatial Indexing: Organizing geometric data for efficient queries

Complexity Considerations

Many geometric algorithms have specific complexity characteristics: - Sweep line algorithms often achieve O(n log n) complexity - Convex hull algorithms typically run in O(n log n) time - Spatial data structures can provide O(log n) query times