Name of Rounded Triangle

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

jewish_scientist
Posts: 903
Joined: Fri Feb 07, 2014 3:15 pm UTC

Name of Rounded Triangle

Postby jewish_scientist » Fri Apr 27, 2018 2:34 pm UTC

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.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5421
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

Re: Name of Rounded Triangle

Postby doogly » Fri Apr 27, 2018 2:47 pm UTC

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.
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?

DavidSh
Posts: 129
Joined: Thu Feb 25, 2016 6:09 pm UTC

Re: Name of Rounded Triangle

Postby DavidSh » Fri Apr 27, 2018 3:27 pm UTC

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.

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5421
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

Re: Name of Rounded Triangle

Postby doogly » Fri Apr 27, 2018 3:39 pm UTC

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?

DavidSh
Posts: 129
Joined: Thu Feb 25, 2016 6:09 pm UTC

Re: Name of Rounded Triangle

Postby DavidSh » Fri Apr 27, 2018 3:59 pm UTC

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.

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5421
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

Re: Name of Rounded Triangle

Postby doogly » Fri Apr 27, 2018 4:29 pm UTC

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?

User avatar
ThirdParty
Posts: 319
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

Re: Name of Rounded Triangle

Postby ThirdParty » Fri May 04, 2018 5:01 am UTC

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
Posts: 903
Joined: Fri Feb 07, 2014 3:15 pm UTC

Re: Name of Rounded Triangle

Postby jewish_scientist » Wed May 09, 2018 2:43 pm UTC

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.

:lol: 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.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

User avatar
ThirdParty
Posts: 319
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

Re: Name of Rounded Triangle

Postby ThirdParty » Wed May 09, 2018 4:51 pm UTC

jewish_scientist wrote: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."


Return to “Mathematics”

Who is online

Users browsing this forum: pex and 10 guests