dimensional object? - 3d geoinformation at tu delft · dimensional object? ken arroyo ohori abe010:...
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How to build an n-dimensional object?
Ken Arroyo Ohori
ABE010: Capita Selecta 20.11.2014
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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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Some background
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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 its 1D boundary
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2D: a face
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2D: a face
which can be described by its 0D boundaries
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2D: a face
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3D: a volume
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3D: a volume
a 3D object can be
described by its 2D boundary
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3D: a volume
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3D: a volume
which can be described by
its 1D boundaries
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3D: a volume
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3D: a volume
which can be described by
their 0D boundaries
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4D, 5D, …
24
= 8 cubes
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however, there is a problem in practice…
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2D: a face
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2D: a face
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2D: a face
a 2D object is described by a set of 1D
objects
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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
same coordinates
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2D: a face
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nextnext
nextnext
next
next
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2D: a face
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nextnext
nextnext
next
next
build an object!
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2D: a face
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3D: a volume
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3D: a volume
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a “soup” of faces
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3D: a volume
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3D: a volume
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build a volume
from a set of faces
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The solution: incremental construction
• Start from a set of 0D vertices
• Connect them to form 1D edges
• Connect these to form 2D faces
• Connect these to form 3D volumes
• …
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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
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2
5
4
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Building two tetrahedra
b
(c)
31
2
5
4
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Building two tetrahedra(d)
d
31
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)
31
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
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Build a tesseract
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build each cube
separately,
then join them
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Build a tesseract
done!
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Methodology• Analyse the problem
• Split it into small, manageable subproblems
• Try sketches/ideas on paper
• When mature enough, build a program to test these
• Start with simple shapes, move towards more complex ones
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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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