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What is the sphere-polyhedron?

Sphere-polyhedron SP (O, p1, p2, ..., pM, r) (Obelongs to setR3, pibelongs to setR3, rbelongs to setR) is a Minkovsky sum

{O}Minkovsky sum l1Minkovsky sum ... Minkovsky sum lM Minkovsky sum S,

where
li = {tpi: -1less or equal t less or equal1}, i = 1, ..., M,
S = {r: |r| less or equal r}.

The point O is called the center of the sphere-polyhedron, the segments li are called the generatrix segments of the sphere-polyhedron, the sphere S is called the generatrix sphere of the sphere-polyhedron. The sphere-polyhedron
g = l1Minkovsky sum ... Minkovsky sum lM Minkovsky sum S = SP (0, p1, p2, ..., pM, r) is called the generatrix set of the sphere-polyhedron; the sphere-polyhedron SP (O, p1, p2, ..., pM, r) we also shall denominate SP(O, g).

Examples:

SP(O, r) (M=0): a sphere with center O and radius r
SP(O, p1, r): a round cylinder with hemispheres on the bottom and on the top
SP(O, i, j, k, 0)
(|i|=|j|=|k|,
iis perpendicular toj, iis perpendicular tok, jis perpendicular tok):		 a cube
SP(O, i, j, k, r): a sphere-cube
SP(O, AB, BC, CD, k, r),
where ABCDEF is a regular
hexagon in the plane ij,
XY=Y-X is a vector
from point X to point Y:		 a sphere-prism

SP(O, p1, p2, ..., pM, 0) is always a center-symmetric polyhedron with center O and center-symmetric facets. In common case (if there are no complanar generatrix segments), all facets are parallelograms.

The polyhedron P0 = SP(O, p1, p2, ..., pM, 0) is called the kernel of any sphere-polyhedron
P = SP(O, p1, p2, ..., pM, r). For example:

Sphere-polyhedron P It's kernel P0

Under Web-browsers Microsoft Internet Explorer 3.01, 4.0 (or later) or Netscape Navigator 3.0, 4.0 (or later), you can build and see any sphere-polyhedron for your generatrix sphere and up to 12 your generatrix segments. To do it, just edit any parameter in a rectangle and leave this rectangle (using mouse or TAB key).

Sphere-polyhedron center
x= , y= , z=
Sphere-polyhedron generatrix set
Radius of the generatrix sphere: R=
Generatrix segments:
1:  gx1= , gy1= , gz1=
2:  gx2= , gy2= , gz2=
3:  gx3= , gy3= , gz3=
4:  gx4= , gy4= , gz4=
5:  gx5= , gy5= , gz5=
6:  gx6= , gy6= , gz6=
7:  gx7= , gy7= , gz7=
8:  gx8= , gy8= , gz8=
9:  gx9= , gy9= , gz9=
10:  gx10= , gy10= , gz10=
11:  gx11= , gy11= , gz11=
12:  gx12= , gy12= , gz12=

(You can set here up to 12 generatrix segments for the sphere-polyhedron. Now there are segments.)

Parameters of the built sphere-polyhedron
Number of vertices: degenerated:
Number of edges: degenerated:
Number of facets:

 

What is the Minkovsky sum?

The Minkovsky sum A Minkovsky sum B of two point set A and B, Ais subset ofR3, Bis subset ofR3, is the set

{(x1+x2, y1+y2, z1+z2): (x1, y1, z1)belongs to setA, (x2, y2, z2)belongs to setB}

For example, if O=(0,0,0), A=(1,0,0), B=(0,1,0), C=(0,0,1) and [XY] denominates the straight segment with ends X and Y, then [OA] Minkovsky sum [OB] Minkovsky sum [OC] is a cube {(x, y, z): 0less or equalxless or equal1, 0less or equalyless or equal1, 0less or equalzless or equal1}.If C=(a,b,0), than [OA] Minkovsky sum [OB] Minkovsky sum [OC] is a plane hexagon (rectangle when a=0 or b=0).