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|43}} | + | {{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|[(xy)(zw)]|43|strange}} |
== Geometry == | == Geometry == |
Revision as of 17:54, 18 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.