Name of Rounded Triangle

[jewish_scientist]

I read about this shape a little while ago, but I forget what it is called. I do remember how to construct it though.

Draw an equilateral triangle on the ground and put a pole at each corner. Now surround the whole thing with a rope that has its ends knotted together. Put a pen inside the rope and move it outward until the rope becomes taut. Move the pen around the triangle, keeping the rope taut. The resulting shape is what I want.

This is not a Reuleaux triangle. A simple proof of this is that a Reuleaux triangle has corners and the shape I am looking for does not.

[doogly]

The Reuleaux triangle is one of these though.

Oh, but I suppose a degenerate case. In general you'd get a hull of three ellipses, not circles.

[DavidSh]

The descriptions of this construction I can find on-line just call it an "Oval", which is a much broader term.
It consists of six elliptical arcs, not three.

[doogly]

Six arcs, from three ellipses.

[DavidSh]

Three sets of foci, but six ellipses.

Let the rope have length L.
Let the poles be A, B, and C, forming a triangle with sides of length d(A,B), d(B,C), d(C,A).
Suppose L is only slightly longer than d(A,B)+d(B,C)+d(C,A).
The elliptical arc near side AB contains points x such that d(x,A)+d(x,B) = L-(d(B,C)+d(C,A)).
The other ellptical arc with these foci is the arc near C. This contains points x such that d(x,A)+d(x,B)=L-d(A,B). These are different ellipses, as long as d(A,B) is different from d(B,C)+d(C,A). That is, as long as the poles A, B, and C aren't co-linear and in order A, C, B. For an equalateral triangle, the ellipses are different.

[doogly]

oh yes, word.

[ThirdParty]

Googled this a bit. Didn't find the answer.

I found a page by Robert Dickau, demonstrating the construction in question with a bunch of very nice gifs. Unfortunately he just calls it "an oval or egg shape".

I found a Wikipedia page on something called a "tri-oval", which included the promising-sounding comment that "according to the four-vertex theorem, every smooth simple closed curve has at least four vertices, points where its curvature reaches a local minimum or maximum. In a tri-oval, there are six such points, alternating between three minima and three maxima". Unfortunately, the rest of the page turned out to be about NASCAR racing.

I also learned about trifocal ellipses, which are not the same shape but are similar in many respects, and Graves's theorem, which uses the same construction method but with a different starting shape. And it occurred to me that while the Reuleaux triangle is not a degenerate example of the shape we're interested in, a standard triangle is: it's the result if the length of the loop of string is equal to the perimeter of the triangle you've looped the string around. (Three of the six ellipse-segments composing the shape we're interested in will have zero length and the other three will have zero curvature.)

[jewish_scientist]

And it occurred to me that while the Reuleaux triangle is not a degenerate example of the shape we're interested in, a standard triangle is: it's the result if the length of the loop of string is equal to the perimeter of the triangle you've looped the string around.

Yeah, it is. I never thought of that. Anyway, the first link provides a citation, so at least I know where to go. Thank you.

[ThirdParty]

Anyway, the first link provides a citation, so at least I know where to go. Thank you.

It's not a terribly useful citation; I followed it before I posted. It went to a math book whose author said "Two readers independently sent me this method for drawing an egg."