Duocylinder (EntityTopic, 14)
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Revision as of 10:55, 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 2D margin of a duocylinder will satisfy the following equations:
x2 + y2 = a2
z2 + w2 = b2
- A duocylinder has two bounding surfaces (3D volumes) which meet at the 2D margin. 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.
- The orthogonal projections of a duocylinder are cylinders. The perspective projections of a duocylinder are intertwined toroidal and hourglass-like shapes.
