A brief and sketchy intro to 3D Convex Hulls

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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

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
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
Representation
• Example
• Incidence graphs
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
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
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})
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
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