1. VORONOI DIAGRAMS FOR PARALLEL HALFLINES IN 3D
Franz Aurenhammer
Institute for Theoretical Computer Science
University of Technology, Graz, Austria

2. n point sites in the plane site p , region reg(p) of
Voronoi diagram
n point sites in the plane
site p , region reg(p) of
points closest to p
Planar straight-line graph
Convex partition of the plane
Linear size, O(n) edges and vertices
Several O(n log n) algorithms
Applications in many areas of science (space partitioning data structure)
Related to many other structures, numerous generalizations.

3. Different sites Line segments, polygons, circular objects…
Still stays linear in size
(planar graph)
But bisectors may be complex
Several construction techniques carry over (D&C, RIC…)
Can be constructed in O(n log n) time

4. Generalized distance functions
Lp norms
Convex distance functions
Weighted Euclidean distances
Geodesic distances
…..
Power function:
Site weights w(p)
(x - p)² - w(p)
Gives the power diagram
Bisectors are lines
Special case: Voronoi diagram, w(p) = 0
O(n log n) time algorithms

5. Three dimensions 3D Voronoi diagram for point sites:
Bisectors are planes
Regions are convex polyhedra
Size varies, from O(n) to quadratic
Well-understood structure
(by duality to convex hulls in 4D)
Many construction algorithms
Worst-case optimal (e.g., insertion method)
Output-sensitive algorithms (via CHs in 4D, O((n+K) log² K) time)

6. Generalized sites in 3D: much more complicated...
Spheres: (relation to PDs) O(n²) size
Lines, line segments: size unclear, Ω(n²)... ~ O(n³)
(envelope bound [Sharir] )
Complex bisectors arise:
point/line, point/plane, line/plane, line/line ...

7. Restricted line segments:
Constantly many orientations: almost quadratic, (2+ε) [Koltun/Sharir]
Nice special case: parallel halflines in 3D
Application: drilling industry (material stability, also geological)
[I. Adamou, PhD Thesis, 2013]
Relatively simple structure
Simple algorithm, computing the combinatorics
Output-sensitive, O(K log n), for size K
Fortunate property: Bisectors behave nicely...
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

8. 3D Straight skeleton (of a polytope)
Bisector properties:
Plane - parabolic cylinder - plane
Intersection with any horizontal plane H is a line!
These lines meet 3-by-3 (as do the 3D bisectors)
so: ∩ with H is a power diagram.
Weights are:
- [(z – zj)+]², z height of H, zj tip of halfline hj
Reveals structural properties
Enables plane-sweep construction in z-direction
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

9. Plane-sweep algorithm:
Start with z < min j zj ( ∩ with H is a Voronoi diagram)
For increasing z: Update the power diagram
(edges of PD... facets of 3D halfline diagram)
Events are:
ai anticipated ends of lifespans of edges ei in the current PD
zj tip of halflines hj
Number of events ai : bounded by the number K of 3D facets
Number of events zj : only n, but involve O(K) updates of lifespans in total
Runtime O(K log K) = O(K log n) … output-sensitive
(no 3D objects involved…)
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

10. 3D Straight skeleton (of a polytope)
Combinatorial size:
Near-quadratic upper bound (O(1) orientations... [Koltun, Sharir])
Is the worst-case size only O(n²) ?
Good properties:
The lowest region is convex. Is its size O(n)?
The intersection of any region with any vertical line is connected.
(Each point in H enters (leaves) a power cell at most once)
Trisector curves are connected and unbounded in z-direction (cycle-free).
(Continuously moving PD vertex in H…)
Bad property:
Two trisector curves can intersect more than once.
(so the O(n) bound for lower envelopes in [Schwartz/Sharir] does not apply)
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

11. 3D Straight skeleton (of a polytope)
Trisector example:
4 halflines given:
(triples define trisectors)
Point on trisector = center of a sphere
that touches 3 halflines
4 halflines can be touched by 2 different
empty spheres [Widmayer]
→ two trisector curves can intersect in 2 points
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

12. Dynamic power diagram:
Sites stay fixed in the sweep-plane H
Weights increase monotonically…
Power edges move self-parallely:
Constant position / constant acceleration / constant speed
O(n²) potential power edges
Can an edge, once having disappeared, come back again?
If not, then each 3D bisector can yield only 1 facet…
3D Straight skeleton (of a polytope)
'Defined' by a similar offsetting process
Polytope facets/edges/vertices trace out the skeleton cells/facets/edges

13. -Thank you-