Duocylinder (EntityTopic, 14)
From Hi.gher. Space
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| - | {{Shape|Duocylinder|''No image''|4|2, 1, 0, 0|0|N/A|N/A|[[Line (object)|E]][[Circle|L]]*EL|22 (x,y),(z,w)|N/A|N/A|N/A}} | + | {{Shape|Duocylinder|''No image''|4|2, 1, 0, 0|0|N/A|N/A|[[Line (object)|E]][[Circle|L]]*EL|22 (x,y),(z,w)|N/A|N/A|N/A|43}} |
== Geometry == | == Geometry == | ||
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</blockquote> | </blockquote> | ||
| - | *A duocylinder has two [[cell]]s which meet at the 2D face. These are given respectively by the systems of equations: | + | *A duocylinder has two [[cell]]s which meet at the 2D face. These are given respectively by the systems of equations:<blockquote> |
| - | + | #''x''<sup>2</sup> + ''y''<sup>2</sup> = ''a''<sup>2</sup>; ''z''<sup>2</sup> + ''w''<sup>2</sup> ≤ ''b''<sup>2</sup> | |
| - | # | + | #''x''<sup>2</sup> + ''y''<sup>2</sup> ≤ ''a''<sup>2</sup>; ''z''<sup>2</sup> + ''w''<sup>2</sup> = ''b''<sup>2</sup> |
| - | + | </blockquote> | |
| + | *Each of these bounding volumes are topologically equivalent to the inside of a 3D torus. The set of points (w,x,y,z) that satisfy either the first or the second set of equations constitute the surface of the duocylinder. | ||
*The [[hypervolume]]s of a duocylinder are given by:{{hmm}} | *The [[hypervolume]]s of a duocylinder are given by:{{hmm}} | ||
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{{Polychora}} | {{Polychora}} | ||
| - | {{ | + | {{Rotope Nav|42|43|44|(((II)I)I)<br>Ditorus|(II)(II)<br>Duocylinder|((II)(II))<br>Tiger}} |
Revision as of 13:07, 17 June 2007
Geometry
A duocylinder is the Cartesian product of two circles.
Equations
- Variables:
a ⇒ radius of the circle in the xy plane
b ⇒ radius of the circle in the zw plane
- All points (x, y, z, w) that lie on the sole 2D face of a duocylinder will satisfy the following equations:
x2 + y2 = a2
z2 + w2 = b2
- A duocylinder has two cells which meet at the 2D face. These are given respectively by the systems of equations:
- x2 + y2 = a2; z2 + w2 ≤ b2
- x2 + y2 ≤ a2; z2 + w2 = b2
- Each of these bounding volumes are topologically equivalent to the inside of a 3D torus. The set of points (w,x,y,z) that satisfy either the first or the second set of equations constitute the surface of the duocylinder.
- The hypervolumes of a duocylinder are given by:(?)
total surface area = 4π2ab
surcell volume = 2π2ab(a + b)
bulk = π2a2b2
- The realmic cross-sections (n) of a duocylinder are cylinders of varying heights.
Projection
The perspective projection of a duocylinder is the following shape. The purple part is one cell, and the black part is the other cell.
http://fusion-global.org/share/duocylinder-04.png
In a parallel projection, both cells collapse to cylinders, one capped and one uncapped, resulting in a single cylinder being observed as the projection.
