A brief and sketchy intro to 3D Convex Hulls Rodrigo Silveira GEOC 2010/11 - Q2

Convex hulls in 3D CH of set of points in 3D: (convex) polytope 2D: CH of n points…. – at most n vertices – at most n edges In 3D it is a bit different – at most n vertices – at most 3n-6 edges – at most 2n-4 facets 2

Representation 2D: CH is a polygon – Easy to store and maintain: vertex array/list 3D: Polytope – More than a list of vertices! – Graph of facets, edges, and vertices 3

Representation Example – Incidence graphs 4

Incremental algorithm Same principle than in 2D – Initialize CH to CH of the first (3+1)=4 points – Incremental step Take next point, p, and insert it Trick: treat points in order by x-coordinate – Then the next point is always outside previous CH – Also works in 2D! Compute CH(P i U {p}) Degeneracy assumption: no 4 points are coplanar 5

Computing CH(P i U {p}) 2D: we add p and 2 edges incident to p 3D: we add p and many facets incident to p 6

The key is in the horizon Horizon: separates visible from non-visible facets Visible facets should be removed Non-visible facets stay in CH(P i U {p}) 7

How to compute them (sketch) Recall points are treated in order The previously inserted point, q, must be in CH(P i ) And must be visible from p Start walking on the facets from q – Depth first search Test each facet to be always on visible part Take note of boundary Connect boundary to p Update graph: delete visible facets, create new vertex, edges, facets 8

Summary This incremental algorithm – In principle, takes time O(n 2 ) – Can be generalized to d dimensions Time O(n log n + n^floor((d+1)/2) Can be adapted to work in expected O(n log n) time 9