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# A brief and sketchy intro to 3D Convex Hulls - PowerPoint PPT Presentation

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

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### A brief and sketchy intro to3D Convex Hulls

Rodrigo Silveira

GEOC 2010/11 - Q2

• 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

• 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

• Example

• Incidence graphs

• 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

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

• Horizon: separates visible from non-visible facets

• Visible facets should be removed

• Non-visible facets stay in CH(Pi U {p})

• 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

• 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