Duocylinder (EntityTopic, 14)
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''b'' ⇒ radius of the circle in the zw plane</blockquote> | ''b'' ⇒ radius of the circle in the zw plane</blockquote> | ||
- | *All points (''x'', ''y'', ''z'', ''w'') that lie on the 2D [[ | + | *All points (''x'', ''y'', ''z'', ''w'') that lie on the sole 2D [[face]] of a duocylinder will satisfy the following equations: |
<blockquote>''x''<sup>2</sup> + ''y''<sup>2</sup> = ''a''<sup>2</sup> <br> | <blockquote>''x''<sup>2</sup> + ''y''<sup>2</sup> = ''a''<sup>2</sup> <br> | ||
''z''<sup>2</sup> + ''w''<sup>2</sup> = ''b''<sup>2</sup> | ''z''<sup>2</sup> + ''w''<sup>2</sup> = ''b''<sup>2</sup> | ||
</blockquote> | </blockquote> | ||
- | *A duocylinder has two | + | *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></blockquote> | #<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></blockquote> | ||
#<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></blockquote> | #<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></blockquote> | ||
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*The [[realmic]] [[cross-section]]s (''n'') of a duocylinder are cylinders of varying heights. | *The [[realmic]] [[cross-section]]s (''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. | ||
+ | <blockquote>http://fusion-global.org/share/duocylinder-04.png</blockquote> | ||
+ | In a parallel projection, both cells collapse to [[cylinder]]s, one [[capped]] and one uncapped, resulting in a single cylinder being observed as the projection. | ||
{{Polychora}} | {{Polychora}} | ||
{{Rotopes}} | {{Rotopes}} |
Revision as of 16:11, 16 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.