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 to 3d convex hulls

A brief and sketchy intro to3D Convex Hulls

Rodrigo Silveira

GEOC 2010/11 - Q2


Convex hulls in 3d
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
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


Representation1
Representation

  • Example

    • Incidence graphs


Incremental algorithm
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 p i u p
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
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
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
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