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.
Name of Rounded Triangle
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Name of Rounded Triangle
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Re: Name of Rounded Triangle
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.
Oh, but I suppose a degenerate case. In general you'd get a hull of three ellipses, not circles.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Name of Rounded Triangle
The descriptions of this construction I can find online just call it an "Oval", which is a much broader term.
It consists of six elliptical arcs, not three.
It consists of six elliptical arcs, not three.
 doogly
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Re: Name of Rounded Triangle
Six arcs, from three ellipses.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Name of Rounded Triangle
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)=Ld(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 colinear and in order A, C, B. For an equalateral triangle, the ellipses are different.
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)=Ld(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 colinear and in order A, C, B. For an equalateral triangle, the ellipses are different.
 doogly
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Re: Name of Rounded Triangle
oh yes, word.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 ThirdParty
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Re: Name of Rounded Triangle
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 "trioval", which included the promisingsounding comment that "according to the fourvertex theorem, every smooth simple closed curve has at least four vertices, points where its curvature reaches a local minimum or maximum. In a trioval, 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 ellipsesegments composing the shape we're interested in will have zero length and the other three will have zero curvature.)
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 "trioval", which included the promisingsounding comment that "according to the fourvertex theorem, every smooth simple closed curve has at least four vertices, points where its curvature reaches a local minimum or maximum. In a trioval, 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 ellipsesegments composing the shape we're interested in will have zero length and the other three will have zero curvature.)

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Re: Name of Rounded Triangle
ThirdParty wrote: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.
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 ThirdParty
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Re: Name of Rounded Triangle
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."jewish_scientist wrote:Anyway, the first link provides a citation, so at least I know where to go. Thank you.
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