Dual Shapes for Convex Polyhedra #1

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1. bdice

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2. b-butler commented on Nov 14, 2019

   

   b-butler

   on Nov 14, 2019

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   @bdice could you flesh this out potentially for the hackathon? I agree that this would be a good feature to have.
3. b-butler

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   good first issueGood for newcomersGood for newcomers

   enhancementNew feature or requestNew feature or request

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4. bdice commented on Nov 16, 2019

   

   bdice

   on Nov 16, 2019

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   @b-butler Here's a starting point. This needs to be tested (with actual tests written) and visually inspected for a few known cases (tetrahedron ↔ tetrahedron, cube ↔ octahedron, dodecahedron ↔ icosahedron). I used the scipy.spatial.HalfspaceIntersection code (see here for docs).

   Sample dual generator script

   import numpy as np
   import scipy.spatial
   
   tetrahedron_vertices = [[0.5, 0.5, 0.5],
                           [-0.5, -0.5, 0.5],
                           [-0.5, 0.5, -0.5],
                           [0.5, -0.5, -0.5]]
   
   hull = scipy.spatial.ConvexHull(tetrahedron_vertices)
   
   print(f'Convex Hull Vertices:\n{hull.points}')
   print(f'Convex Hull Volume: {hull.volume}')
   print(f'Convex Hull Equations:\n{hull.equations}')
   
   dual = scipy.spatial.HalfspaceIntersection(
       halfspaces=hull.equations, interior_point=np.zeros(3))
   
   print(f'Dual Vertices:\n{dual.dual_points}')
   print(f'Dual Surface Area: {dual.dual_area}')
   print(f'Dual Volume: {dual.dual_volume}')
   print(f'Dual Equations:\n{dual.dual_equations}')
   print(f'Dual Facets: {dual.dual_facets}')

   Sample dual output

   Convex Hull Vertices:
   [[ 0.5  0.5  0.5]
    [-0.5 -0.5  0.5]
    [-0.5  0.5 -0.5]
    [ 0.5 -0.5 -0.5]]
   Convex Hull Volume: 0.3333333333333334
   Convex Hull Equations:
   [[-0.57735027  0.57735027  0.57735027 -0.28867513]
    [ 0.57735027 -0.57735027  0.57735027 -0.28867513]
    [-0.57735027 -0.57735027 -0.57735027 -0.28867513]
    [ 0.57735027  0.57735027 -0.57735027 -0.28867513]]
   Dual Vertices:
   [[-2.  2.  2.]
    [ 2. -2.  2.]
    [-2. -2. -2.]
    [ 2.  2. -2.]]
   Dual Surface Area: 55.425625842204084
   Dual Volume: 21.33333333333334
   Dual Equations:
   [[-0.57735027 -0.57735027  0.57735027 -1.15470054]
    [ 0.57735027  0.57735027  0.57735027 -1.15470054]
    [-0.57735027  0.57735027 -0.57735027 -1.15470054]
    [ 0.57735027 -0.57735027 -0.57735027 -1.15470054]]
   Dual Facets: [[2, 1, 0], [3, 1, 0], [3, 2, 0], [3, 2, 1]]

   Additional references:

   • https://www.georgehart.com/virtual-polyhedra/duality.html
   • https://en.wikipedia.org/wiki/Dual_polyhedron
   • http://mathworld.wolfram.com/DualPolyhedron.html
5. b-butler commented on Nov 16, 2019

   

   b-butler

   on Nov 16, 2019

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   @bdice thanks!
6. janbridley

   self-assigned this

   on Apr 10, 2023