Duocylinder (EntityTopic, 14)

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Revision as of 16:11, 16 June 2007 by Keiji (Talk | contribs)

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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.

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