The Discrepancy Problem Point Continuous measure Discrete measure
The Discrepancy Problem Point Continuous measure: Discrete measure: exactly one point (Type i) brute-force method at least two points (Type ii) apply duality + line arrangement
Arrangement of Lines Point with unbounded edges and faces Simple arrangement if no three lines are concurrent; no two lines are parallel. face ge d e vertex
Reduction to Line Arrangment Point Problem on points problem on an arrangement of dual lines. primal plane dual plane Structure of a line arrangement is more apparent than that of a point set.
Combinatorial Complexity #vertices + #edges + #faces Point
Proof of Complexity Point small enough rotation to yield new vertex Complexity increases in this case. (Hence such configuration cannot be maximal. )
Proof (Cont’d) Point slightly 2 new vertices 3 new edges 1 new face Since complexity increases, such configuration cannot be maximal either. The arrangement with maximal complexity must be simple.
Exact Size of a Simple Alignment # vertices Point # edges # faces Any pair of lines intersect. #edges on one line = 1 + #intersections on the line
Number of Faces Point No edge crossing Planar graph Euler’s formula:
Storage of Line Arrangement Doubly-connected edge list. Point Add a bounding box to contain all vertices in interior. (marks the last slide in this video segment)
Plane sweep? Point pairwise intersection Not optimal!
Incremental Algorithm Preprocessing Point leftmost, rightmost, top, bottom intersections. Update the DCEL after each addition.
Updating the Subdivision Point
Updating the Subdivision Point Alternatively use Next() and Twin() pointers.
First Edge of Intersection Point
Splitting a Face Point 2 new faces 1 new vertex 6 new half-edges new vertex
The Algorithm Point
Face Splitting Point Total complexity?
Zone Point
Time of Arrangement Construction Point Proof By induction. Time to insert all lines, and thus to construct line arrangement:
Discrete Measure Primal plane Point Dual plane Efficient algorithm exists!
How to Use Duality? Point
Reduction Point
Levels of Vertices in an Arrangement level of a point = # lines strictly above it. Point 0 1 2 2 3 3 4 1 2 3 3
Counting Levels of Vertices
Counting Levels of Vertices no change of level Point between vertices 1 1 0 0 0 1 2 2 1 3 coming from above coming from below
Running Times Point
Discrete Measures & Degeneracy Point
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