Assess your understanding of key computational geometry algorithms, concepts, and applications including convex hulls, intersection tests, Voronoi diagrams, and polygon operations. Gain insights into solving geometric problems critical for computer graphics, robotics, and spatial analysis.
Which algorithm efficiently computes the convex hull of a set of points in two dimensions and is known for its O(n log n) time complexity?
Explanation: Graham Scan is a well-known algorithm for finding the convex hull of a set of points in the plane with O(n log n) time complexity. Dijkstra's Algorithm is used for shortest paths in graphs, not for convex hulls. Ford-Fulkerson solves the maximum flow problem. Hamming Sort is not a recognized computational geometry algorithm. Thus, Graham Scan is the correct answer here.
When checking if two line segments in a plane intersect, which initial check can significantly reduce computational effort before performing exact intersection tests?
Explanation: Checking if the bounding boxes of the segments overlap is a fast way to rule out many non-intersecting pairs before computing the exact intersection. Color similarity is irrelevant to geometry. Neighbor list lookup applies to graphs or meshes, not individual segments. Dot product calculation alone does not determine segment intersection. Therefore, bounding box overlap is the most effective preliminary check.
Which geometric structure divides the plane into regions around a set of points so that each region consists of all points closer to one particular point than to any other?
Explanation: A Voronoi Diagram partitions the plane according to the nearest point principle, creating convex polygonal regions for each given site. A Minimum Spanning Tree connects nodes with minimum total edge length but does not partition space. A Binary Search Tree is a data structure unrelated to plane divisions. Delaunay Heap is not a geometric structure. Thus, Voronoi Diagram is correct.
Given the vertices of a simple (non-intersecting) polygon listed in order, which algorithm efficiently computes the area in linear time?
Explanation: The Shoelace Formula computes the area of a simple polygon by summing ordered vertex products, operating in linear time relative to the number of vertices. Kruskal's Algorithm is used for minimum spanning trees, not polygons. Sutherland-Hodgman is for polygon clipping, not area calculation. Fast Marching Method solves boundary value problems, not polygon areas. Therefore, Shoelace Formula is the right choice.
In computational geometry, which algorithm is commonly used to decide if a given point lies inside, outside, or on the boundary of a polygon?
Explanation: The Ray Casting Method counts edge crossings by a ray from the point to determine its positional relationship with the polygon. Two-Phase Commit is a database protocol. Polar Sieve is not a standard geometric algorithm. Alpha-Beta Pruning is used in game tree search, not geometric containment. Thus, Ray Casting Method is the correct answer.