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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
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