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

The Algorithm Point

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

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

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

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

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

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*).

Levels of Vertices in an Arrangement Point level of a point = # lines strictly above it.

Counting Levels of Vertices Point

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 no change of level between vertices

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.

Duality in Higher Dimensions Point point hyperplane point

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

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.

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