Nonoverlap of the Star Unfolding

1. Nonoverlap of the Star Unfolding Boris Aronov and Joseph O’Rourke, 1991 A Summary by Brendan Lucier, 2004

2. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

3. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

4. Introduction – What is this paper about, and why do I like it? • It is a mathematics paper with algorithmic consequences • Proves theorems and shows how they simplify existing algorithms • Single-source shortest path queries (Chen and Han, 1990) • Finding edge sequences for shortest paths (Sharir and Schorr, 1986) • Computing geodesic diameter of a polyhedron (Agarwal, Aronov, O’Rourke, and Schevon, 1990)

5. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

6. Shortest Paths on a Polyhedron • We are given a convex polyhedron P. We shall refer to the vertices of P as corners. • Given points x and y on a polyhedron P, we shall say “the shortest path from x to y” to mean the shortest path lying on the surface of P. • Note that the shortest path need not be unique.

7. Ridge Trees • Fix a point x on P. We shall call x the source point. • We require that the shortest path from x to each corner be unique. • A point y is a Ridge Point if there are multiple distinct shortest paths from x to y. • The set of all ridge points forms a tree (not obvious) called the Ridge Tree. A vertex of this tree is a Ridge Vertex. • Note that the ridge tree touches each corner (exercise: prove it).

8. The Star Unfolding • Take a convex polyhedron P with n corners. • Choose a point x on P with a unique shortest path to each corner of P. • Cut P along the shortest paths from x to each corner. Note that there will be n cuts. • Unfold the polyhedron into the plane. • This is a Star Unfolding Sx of P.

9. Example: Square Pyramid b c a A pyramid with source point and shortest paths to corners [a]. The star unfolding with edge associations [b] and with the original polyhedron edges [c].

10. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

11. Basic Properties of Star Unfoldings • Sx has 2n edges; two for each cut made in the polyhedron. • Pairs of corresponding edges are adjacent.

12. Basic Properties (con’t) • The vertices of Sx are the images of x and the n corners of P. • Each corner has exactly one image in Sx. • The other n vertices are all images of x. • Each cut connects x to a corner, so each edge must connect an image of x to an image of a corner.

13. Star Unfolding with Ridge Tree • We now add the ridge tree to the representation of Sx. • “Paint” the ridge tree onto the polyhedron, then unfold. • Notice that the ridge tree looks like a Voronoi diagram for the source points xi. More on this later.

14. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

15. Outline of Main Theorem • Theorem: The star unfolding Sx of any convex polyhedron does not self-overlap. • We proceed by induction on n, the number of corners of P. • The proof is split into the following steps: • Give a reduction from the star unfolding Sx, a polygon with 2n sides, to a new polygon Sx’ with 2(n-1) sides. • Show that Sx’ is the star unfolding of a convex polyhedron. • Prove that non-overlap of Sx follows from non-overlap of Sx’. • Handle the base case.

16. Step 1: The Reduction • Lemma: there is a ridge vertex v adjacent to two consecutive corners pi, pi+1, whose sum of curvatures is at most 2π. • Take the hexagon illustrated. Remove pi and pi+1 and replace them with p’. We position p’ so its external angle equals the sum of the curvatures of pi and pi+1. • The resulting polygon is Sx’.

17. Reduction – Special Case • If n=4, then the sum of curvatures at any two corners could be 2π (e.g. a regular tetrahedron). • It’s difficult to create point p’ with exterior angle 2π! • We handle this by placing p’ “at infinity,” so the polygon is infinite. • This case is handled separately. a b c A tetrahedron with source point and shortest paths [a], the star unfolding with polyhedron edges shaded in gray [b], and the reduction to an infinite polygon (interior shaded) [c].

18. Step 2: Show S’ is a Star Unfolding • Alexsandrov’s Theorem: Every net that is homeomorphic to a sphere and whose angle sum at every vertex is ≤ 2π corresponds to a closed convex polyhedron. • The authors show that Alexsandrov’s Theorem applies to Sx’ (not too hard), so Sx’ is an unfolding of some polyhedron P’. • The paper also proves that Sx’ is actually the star unfolding of P’. S S’

19. Step 3: Nonoverlap • Want to claim that Sx does not overlap if Sx’ does not overlap. • Problem: the edges in Sx can go “outside” the edges in Sx’. • Idea: Expand the unfolding to include “Sectors.” • A Sector is the exterior sector of the circle centred at a corner pi, with points xi and xi-1 on its circumference.

20. Example: Pyramid Unfolding

21. Step 3 (con’t) • The authors show that Sx plus sectors is contained in Sx’ plus sectors. • It then follows that if the “big” sector of Sx’ doesn’t intersect any other part of Sx’, then the two “small” sectors of Sx won’t either. • This concludes the induction step.

22. Step 4: Base Case • The base case occurs when n=3. • This is not a polyhedron, but a “doubly-covered triangle” that is allowed by Alexsandrov’s Theorem. • The proof of the claim is “easy” for the base case, but too technical to get into here.

23. Outline • Introduction • Definitions • Basic Properties • Main Theorem • Algorithmic Consequences

24. Voronoi Property • Let X be the set of images of x in the star unfolding. • Theorem: the ridge tree is exactly the Voronoi diagram of X, intersected with the star unfolding. • A non-trivial corollary to the induction argument. • The main idea is to view the sectors as Voronoi Disks.

25. Consequences of the Voronoi Property • It is now easy to construct the ridge tree: • Generate the star unfolding Sx. • Compute the Voronoi diagram V of X. • Restrict V to the interior of Sx. • This can be applied to previous algorithms: • Justifies “Single-source shortest path queries” (Chen and Han, 1990) • Simplifies the analysis of “Finding edge sequences for shortest paths” (Sharir and Schorr, 1986) • Improves running time of “Computing geodesic diameter of a polyhedron” (Agarwal, Aronov, O’Rourke, and Schevon, 1990)

26. Conclusion: This Paper… • developed some properties of Ridge Trees and Star Unfoldings, • proved that the Star Unfolding of a convex polyhedron does not self-overlap, and • applied this theorem to existing algorithms. Fin.