Reuleaux Triangle
The Reuleaux Triangle is a set of constant width. That means, if you measure its width with calipers, you'll get the same measurement no matter how it's oriented--but it isn't a circle!
In this applet, there are two frames. On the left, we have a Reuleaux triangle with fixed center at (0,0). You can turn it by moving point B. Notice how the bounding box moves, but doesn't change size.
Point D is the center of the bounding box. If you full-screen the applet, you may notice that two curves are constructed through D: one is the circle about (0,0) through D; the other is the trajectory of D as the triangle turns. They aren't the same curve. (If you watch closely, you can see the circle change size in order to stay in touch with D; or, you might just notice that one of the curves is more square than the other.)
On the right, we have a Reuleaux triangle whose center wobbles according to the movement of point D in order to keep a fixed bounding box. The three vertices of this Reuleaux triangle sweep out an interesting curve. I wonder: why are the corners rounded? Could you fix it?