Duocylinder (EntityTopic, 14)
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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|[(xy)(zw)]|43|strange}} | {{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}} | ||
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A '''duocylinder''' is the [[Cartesian product]] of two [[circle]]s. It is also the limit of the [[set]] of [[duoprism]]s as ''m'' and ''n'' tend to infinity. | A '''duocylinder''' is the [[Cartesian product]] of two [[circle]]s. It is also the limit of the [[set]] of [[duoprism]]s as ''m'' and ''n'' tend to infinity. | ||
| - | + | == Equations == | |
*Variables: | *Variables: | ||
<blockquote>''a'' ⇒ radius of the circle in the xy plane<br> | <blockquote>''a'' ⇒ radius of the circle in the xy plane<br> | ||
Revision as of 20:24, 22 September 2007
Template:Shape A duocylinder is the Cartesian product of two circles. It is also the limit of the set of duoprisms as m and n tend to infinity.
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.
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
| 42. [(xy)<zw>] Narrow cubinder | 43. [(xy)(zw)] Duocylinder | 44. ExPar: unexpected closing bracket Large hexadecachoron |
| List of bracketopes | ||
