Computational Geometry

9.

Arrangements and Duality

9.1.

Line Arrangements

9.1.1.

Arrangement Definition

9.1.1.1.

9.1.1.2.

9.1.1.3.

9.1.2.

Combinatorial Complexity

9.1.2.1.

9.1.2.2.

9.1.2.3.

9.1.2.4.

Euler's Formula

9.1.3.

Construction Algorithms

9.1.3.1.

Incremental Construction

9.1.3.1.1.

9.1.3.1.2.

9.1.3.1.3.

Complexity Analysis

9.1.3.2.

Divide and Conquer

9.1.3.2.1.

Recursive Construction

9.1.3.2.2.

Merging Arrangements

9.1.4.

9.1.4.1.

9.1.4.2.

Proof Techniques

9.1.5.

Single Cell Complexity

9.1.6.

Many Faces Problem

9.2.

Duality Transform

9.2.1.

Point-Line Duality

9.2.1.1.

Transformation Rules

9.2.1.2.

Incidence Preservation

9.2.1.3.

Order Preservation

9.2.2.

Properties of Duality

9.2.2.1.

Involution Property

9.2.2.2.

Geometric Interpretations

9.2.3.

9.2.3.1.

Ham Sandwich Theorem

9.2.3.2.

Linear Programming

9.2.3.3.

Range Searching

9.2.4.

Projective Duality

9.3.

Hyperplane Arrangements

9.3.1.

Higher-Dimensional Arrangements

9.3.2.

Complexity Bounds

9.3.3.

Construction Methods

9.3.4.

9.3.4.1.

Linear Programming

9.3.4.2.

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10. Motion Planning