A brief and sketchy intro to3D Convex Hulls

Rodrigo SilveiraGEOC 2010/11 - Q2

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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

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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

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Representation

• Example– Incidence graphs

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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(Pi U {p})

• Degeneracy assumption: no 4 points are coplanar

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Computing CH(Pi U {p})

• 2D: we add p and 2 edges incident to p• 3D: we add p and many facets incident to p

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The key is in the horizon

• Horizon: separates visible from non-visible facets• Visible facets should be removed• Non-visible facets stay in CH(Pi U {p})

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How to compute them (sketch)• Recall points are treated in order• The previously inserted point, q,

must be in CH(Pi)• 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

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Summary

• This incremental algorithm– In principle, takes time O(n2)– 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