Duocylinder (EntityTopic, 14)

From Hi.gher. Space

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''b'' ⇒ radius of the circle in the zw plane</blockquote>
''b'' ⇒ radius of the circle in the zw plane</blockquote>
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the 2D [[margin]] of a duocylinder will satisfy the following equations:
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*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>
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*A duocylinder has two bounding surfaces (3D volumes) which meet at the 2D margin. These are given respectively by the systems of equations:
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*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> &le; ''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> &le; ''b''<sup>2</sup></blockquote>
#<blockquote>''x''<sup>2</sup> + ''y''<sup>2</sup> &le; ''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> &le; ''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.
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*The orthogonal projections of a duocylinder are cylinders. The perspective projections of a duocylinder are intertwined toroidal and hourglass-like shapes.
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== Projection ==
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The perspective projection of a duocylinder is the following shape. The purple part is one cell, and the black part is the other cell.
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<blockquote>http://fusion-global.org/share/duocylinder-04.png</blockquote>
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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

Template:Shape

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:
  1. x2 + y2 = a2; z2 + w2b2
  2. x2 + y2a2; 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.
total surface area = 4π2ab
surcell volume = 2π2ab(a + b)
bulk = π2a2b2

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.

Template:Polychora Template:Rotopes