1. Splitting a Face Point edge already split new vertex two new faces one new vertex six new half-edges

2. The Algorithm Point

3. Face Splitting Point Split faces in intersected by. - Insertion of takes time linear in total complexity of faces intersected by the line.

4. Zone Point once three times A vertex may be counted up to four times. four times twice

5. Time of Arrangement Construction Point Proof By induction. Time to insert all lines, and thus to construct line arrangement:

6. Solution to the Discrepancy Problem Point Continuous measure: Discrete measure: Minimize Exactly one point  brute-force method At least two points  apply duality

7. How to Use Duality? Point A line through ≥ 2 sample points

8. Reduction Point # lines above a vertex # lines below the vertex # lines through the vertex Sufficient to compute 2 of 3 numbers (with sum n). is known from DCEL. Need only compute the level of every vertex in A(S*).

9. Levels of Vertices in an Arrangement Point 0 2 1 3 1 3 4 3 2 3 2 level of a point = # lines strictly above it.

10. Counting Levels of Vertices Point

11. Counting Levels of Vertices Point 1 Along the line the level changes only at a vertex. A line crossing comes either from above or from below (relative to the current traversal position) a) from above  level( ) = level( ) – 1b) from below  level( ) = level( ) + 1 coming from above coming from below 0 1 1 3 1 0 0 22 no change of level between vertices

12. Running Times Point Levels of all vertices in a line arrangement can be computed in time. Discrete measures computable in time. Levels of vertices along a line computable in time.

13. Duality in Higher Dimensions Point point hyperplane point

14. Inversion Point point The point is lifted to the unit paraboloid.

15. Image of a Circle Point Image of a point on the circle C has z-coordinate plane is the intersection of the plane P with the unit paraboloid.

16. Inside/Outside  Below/Above Point lies inside C iff is below.