#geometry
CF 1949A - Grove
CF 1949A - Grove Rating: 3300 Tags: brute force, dfs and similar, dp, geometry, probabilities Solve time: 2m 41s Verified: yes Solution Problem Understanding Every tree is planted at an integer lattice point. Around that point we place a disk of radius r , representing the root system. Two conditions must hold. The entire disk must stay inside the square lawn. Since the lawn is the square [0,n] × [0,n]...
CF 241C - Mirror Box
CF 241C - Mirror Box Rating: 2000 Tags: geometry, implementation Solve time: 1m 27s Verified: no Solution Problem Understanding The system describes a rectangular box where a laser beam enters through one small hole on the left wall and must exit through another hole on the right wall. Inside the box, there are horizontal mirror segments placed either on the floor or on the ceiling. Each mirror covers a segment...
CF 46G - Emperor's Problem
CF 46G - Emperor's Problem Rating: 2500 Tags: geometry Solve time: 1m 33s Verified: no Solution Problem Understanding We are asked to construct a convex polygon with $n$ vertices that satisfies three conditions. First, all vertices must lie on lattice points, meaning each coordinate is an integer. Second, all sides must have distinct lengths. Third, among all polygons that satisfy these two conditions, the one with the minimal possible longest...
52. Convex and Discrete Geometry
This volume studies convex sets, polytopes, and discrete geometric structures. It emphasizes combinatorial structure, geometric inequalities, and computational aspects. Part I. Convex Sets Chapter 1. Convexity 1.1 Convex sets and combinations 1.2 Convex hulls 1.3 Extreme points 1.4 Carathéodory theorem 1.5 Examples Chapter 2. Separation and Support 2.1 Supporting hyperplanes 2.2 Separation theorems 2.3 Dual cones 2.4 Applications 2.5 Examples Chapter 3. Convex Functions 3.1 Definitions 3.2 Properties 3.3 Jensen...
51. Geometry
This volume studies geometric structures, transformations, and invariants. It covers classical geometry, synthetic methods, and modern structural viewpoints. Part I. Foundations of Geometry Chapter 1. Geometric Objects 1.1 Points, lines, planes 1.2 Incidence and betweenness 1.3 Distance and angle 1.4 Congruence and similarity 1.5 Examples Chapter 2. Axiomatic Geometry 2.1 Euclidean axioms 2.2 Hilbert-style axioms 2.3 Models of geometry 2.4 Independence of axioms 2.5 Examples Chapter 3. Transformations 3.1 Isometries...
53. Differential Geometry
This volume studies smooth geometric structures using calculus. It develops curves, surfaces, manifolds, and curvature, forming the foundation for modern geometry and physics. Part I. Curves and Surfaces Chapter 1. Curves in Euclidean Space 1.1 Parametrized curves 1.2 Arc length 1.3 Curvature and torsion 1.4 Frenet frame 1.5 Examples Chapter 2. Surfaces 2.1 Parametrized surfaces 2.2 Tangent planes 2.3 First fundamental form 2.4 Surface area 2.5 Examples Chapter 3. Curvature...
3.4 Geometric Data Structures
3.4 Geometric data structures, 40 index slug name 1 geometric-data-structure Geometric Data Structure 2 point-set Point Set 3 line-segment-set Line Segment Set 4 interval-tree-geometric Interval Tree 5 segment-tree-geometric Geometric Segment Tree 6 range-tree-geometric Range Tree 7 two-dimensional-range-tree-geometric 2D Range Tree 8 kd-tree KD Tree 9 kd-tree-nearest-neighbor KD Tree Nearest Neighbor 10 ball-tree Ball Tree 11 vp-tree Vantage Point Tree 12 cover-tree Cover Tree 13 quadtree Quadtree 14 octree Octree 15...