Information Technology Reference
In-Depth Information
(a)
(b)
(c)
(d)
Fig. 1. Examples of imaginary cubes: (a) regular tetrahedron, (b) cuboctahedron, (c)
H: hexagonal bipyramid with 12 isosceles triangle faces with a height 3/2 of the base,
(d) T: triangular antiprismoid obtained by t runcating the three vertices of a base of a
regular triangular prism whose height is 6 / 4ofanedge.
(a)
(b)
(c)
(d)
Fig. 2.
Imaginary cubes in Fig. 1 placed in cubes.
face of P . We simply call an n -dimensional hypercube an n - cube . We refer the
reader to [ 5 ] for background material on polytopes.
For any two objects A and B , and for any scalar c
R
, we set their Minkowski
n
sum A + B =
{ a
+
b R
| a
A,
b
B
}
, and scaling cA =
{
c
a | a
A
}
.In
n , and “
n .
this paper, 1 is the vector (1 ,..., 1)
R
·
” is the dot product on
R
2
Imaginary Cubes
Imaginary cubes are three-dimensional objects with square projections in three
orthogonal ways. Note that a regular octahedron also has square projections in
three orthogonal ways, but its square projections are arranged differently. We
exclude such a case by defining an imaginary cube more precisely as follows.
Definition 1. Let C be a 3-cube, and A be an object.
1. A is an imaginary cube of C if A has the same three square projections as
C has.
2. A is an imaginary cube if it is an imaginary cube of a cube.
3. A is a minimal convex imaginary cube (MCI for short) of C if A is minimal
among convex imaginary cubes of C .
4. A is an MCI if it is an MCI of a cube.
It is clear that a convex object A is an imaginary cube of C if and only if
each edge of C contains at least one point of A . Therefore, an MCI of C is a
convex hull of some points of the edges of C , and thus it is a polytope.
 
Search WWH ::




Custom Search