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Incremental construction of nD objects
Ken Arroyo Ohori
GeoBuzz ’s-Hertogenbosch
25.11.2014
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The 24-cella “simple” 4D object
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The 24-cell
24 0D vertices 96 1D edges 96 2D faces
24 3D volumes 1 4D hypervolume
a “simple” 4D object
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objects in more than 3D are complex!
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Defining 0D-3D objects in nD space
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0D: a vertex
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0D: a vertex
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1D: an edge
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1D: an edge
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1D: an edge
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1D: an edge
a 1D object can be described by its 0D boundaries
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2D: a face
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2D: a face
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2D: a face
a 2D object can be described by
a set of 1D objects that form
its boundary
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2D: a face
a “soup” of line segments
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2D: a face
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2D: a face
find common 0D vertices on the boundary of a set of 1D
edges
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2D: a face
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nextnext
nextnext
next
next
2D: a face
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nextnext
nextnext
next
next
generate topological
relationships
2D: a face
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eliminate redundant elements
3D: a volume
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3D: a volume
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3D: a volume
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a “soup” of faces
3D: a volume
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3D: a volume
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find common 1D edges
on the boundary of a set of 2D
faces
Incremental construction• Start from a set of 0D vertices
• Connect them to create 1D edges
• Connect 1D edges, forming 2D faces by finding common 0D vertices
• Connect 2D faces, forming 3D volumes by finding common 1D edges
• Connect nD cells, forming (n+1)D cells by finding common (n-1)D cells
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Building two tetrahedra
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Building two tetrahedra(a) 1
2
3
4
5
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Building two tetrahedra(b)
a
31
2
5
4
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Building two tetrahedra
b
(c)
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2
5
4
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Building two tetrahedra(d)
d
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2
5
4
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Building two tetrahedra(e)
c
31
2
5
4
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Building two tetrahedra
e
(f)
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2
5
4
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Building two tetrahedra(g)
f
31
2
5
4
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Building two tetrahedra
g
(h)
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Building two tetrahedra
c
a
ge
fd
b
(a)
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Building two tetrahedra
c
a
ge
f
b
d
(b)
done!
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Build a tesseract
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Build a tesseract
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build each cube
separately
Build a tesseract
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build each cube
separately,
then join them
Build a tesseract
done!
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Images from:
• http://commons.wikimedia.org/wiki/File:Stereographic_polytope_24cell_faces.png
• http://blogs.lt.vt.edu/foundationdesignlab/category/materials/
• http://commons.wikimedia.org/wiki/File:Schlegel_wireframe_8-cell.png
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More info
Ken Arroyo Ohori, Guillaume Damiand and Hugo Ledoux. Constructing an n-dimensional cell complex from a soup of (n-1)-dimensional faces. In Prosenjit Gupta and Christos Zaroliagis (eds.), Applied Algorithms, Volume 8321 of Lecture Notes in Computer Science, Springer International Publishing Switzerland, January 2014, pp. 37–48.
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