# Help me visualize this geometric solid!

Not that exciting a topic, so I thought an explanation point would help.

This post is inspired by a homework problem, but I did my work and solved the problem (finding the volume), so it’s kosher to post. I’m having the most difficult time figuring out what this thing would look like:

A circle is the base of a solid. Each cross-section perpendicular to the x-axis is a square.

So in other words, you have a circle flat on your desk, and at any point, the height of the solid is the same as the length of a chord going through that point, perpendicular to the x-axis.

It’s not a cone, or a cylinder, or any other typical shape. Maybe more like half a football… I think. Anyone?

Exclamation point, even.
(!)

If you found the volume, there should be equations, yes? Those would help for visualization.

If you found the volume, there should be equations, yes? Those would help for visualization.

I’m pretty far removed from being a geometry wiz, but that sounds like it would be cylindrical to me.

Questions like these are the reason God invented Play-Doh. Seriously, I’ve got a couple cans in my desk for just such situations.

The resulting shape has three edges. The first is the circular base. The other two are two symmetrical edges that start at a single point on the circle, rise and spread apart (staying directly above the circle, natch), then fall back to meet at a second point 180 degrees from the first.

You can get some idea of the shape by looking here (a 1/4-circle rather than full-circle base, but same general idea).

I made a 3d solid of the shape, you can find a pic of it here
What can I say, I really seem to have too much time.

If the circular base lies in the xy-plane and is centered at the origin, then the length of a chord of the circle parallel to the y-axis and lying a distance x from the y-axis is just 2sqrt(R[sup]2[/sup] - x[sup]2[/sup]). This means that the height of the “roof” as a function of x is also 2sqrt(R[sup]2[/sup] - x[sup]2[/sup]), and so the solid is bounded by three surfaces:

z = 0 (the bottom face)
x[sup]2[/sup] + y[sup]2[/sup] = R[sup]2[/sup] (the outside of the cylinder)
z = 2*sqrt(R[sup]2[/sup] - x[sup]2[/sup]) (the “roof”)

Note that the last equation can be re-written as

z[sup]2[/sup]/4 + x[sup]2[/sup] = R[sup]2[/sup]

which is the equation for an ellipse centered at the origin in the xz-plane. So if you looked at the object from along the y-axis, it would look like half of an ellipse with principal axes of lengths R and 2R.

I’ll see if I can scare up some Mathematica plots for you.

But if you intersect this object with a plane perpendicular to the X-axis would the “slice” be square?

Here’s a plot. (Hope I’m not stepping on your toes, missing_link, but I couldn’t get your link to load.)

ccwaterback: Yeah, I think that’s how I defined it. My elliptical (hah!) comment above was talking about taking the slice y=0, i.e. a slice perpendicular to the y-axis, instead.

Cool, very nice.

I kept getting stuck thinking you have to be able to rotate this beast about the y-axis and still come up with squares.

There are several different shapes shown. I think the one we want is

http://astro.temple.edu/~dhill001/sectionmethod/sqrcross75slab.gif

MikeS, your link looks right to me as well.

Seifert…

You’re not related to the Seifert, are you?

Ah. This is fabulous. Particularly this:

And the plot. I love The Dope.

You mean the topologist? Not to the best of my knowledge, but anything’s possible. My branch of the Seiferts came to North America four or five generations ago, so I’m definitely not directly descended.

Ah, oh well. That would have been cool, though. My colleagues would have enjoyed it at least.