tapertop and duocylinder

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tapertop and duocylinder

Postby papernuke » Fri Sep 22, 2006 3:37 am

i still dont get what a tapertop and duocylinder is (dont tell me to look it up i tried that on all the search machines i know already).

[edit] don't tell me with notations, i don't understand any of them

[edit (again)] equations*
Last edited by papernuke on Mon Sep 25, 2006 4:04 am, edited 1 time in total.
"Civilization is a race between education and catastrophe."
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Postby pat » Fri Sep 22, 2006 4:00 am

Consider a normal, three-dimensional cylinder. What can you do to make it four-dimensional?

Let's think about how we made the three-dimensional cylinder. You could think about starting with a rectangle and spinning on an axis through the center and parallel to one of the sides. You could think about starting with a disc and stretching it up to three dimensions.

I'm going to ask you to think about a different way. Assume we started with a solid, 3-dimensional cube of wood. We put it on a lathe. We mount it so that it will spin on an axis that goes through the center of two opposite faces of the cube. Now, we lathe it down to get a cylinder.

If we had instead started with a 4-dimensional cube of wood and put it on a lathe, we could do the same thing. The spinning of the lathe will leave two-dimensions fixed while rotating the other two dimensions. When we're through, we would have the cubinder.

Now, we're going to take the cubinder off of the lathe and then put it back on in a different orientation. The orientation that we're going to pick to remount it will take both of the dimensions that were fixed before and rotate them. At the same time, the dimensions that rotated before will stay fixed. Lathe the cubinder down in this way, and you've got a duocylinder.

If you found the second lathing confusing, another way would be to make two cubinders. Turn one of them so that its flat edges are perpendicular to the flat edges of the other. Now, mash them together keeping only the parts where they overlap. This intersection of two cubinders is a duocylinder (and, is how I do duocylinders in my raytracer).
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Postby pat » Fri Sep 22, 2006 4:04 am

The top rows in this picture are, respectively: tesseract, cubinder, spherinder, sphere, and duocylinder.

Image
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