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Subject: "Reuleaux's triangle"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #287
Reading Topic #287
David
guest
May-29-04, 08:20 PM (EST)
 
"Reuleaux's triangle"
 
   When Reuleaux's triangle is rotated in a square, what shape does it's center trace? What shape do the set of midpoints of all diamters of Reuleaux's triangle make?


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alexb
Charter Member
1318 posts
May-29-04, 08:28 PM (EST)
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1. "RE: Reuleaux's triangle"
In response to message #0
 
   >When Reuleaux's triangle is rotated in a square, what shape
>does it's center trace?

This shape consists of four elliptic arcs. (see https://mathworld.wolfram.com/ReuleauxTriangle.html with a reference to S. Wagon's Mathematica in Action, New York: W. H. Freeman, pp. 52-54 and 381-383, 1991

>What shape do the set of midpoints
>of all diamters of Reuleaux's triangle make?

In the above mentioned book there may be something about that as well.


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sfwc
Member since Jun-19-03
May-31-04, 10:27 AM (EST)
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2. "RE: Reuleaux's triangle"
In response to message #0
 
   Well, since the first question has been adequately answered, I'll consider the second. A diameter of Reuleaux's triangle must be a line joining a vertex to a point on the opposite arc. So the locus of such points has 3 sections, 1 for each vertex. Now, for each vertex, the locus is the locus of midpoints of lines joining that vertex to the opposite arc, so it is the opposite arc scaled by a factor of two about the vertex. So the complete locus consists of 3 60 degree arcs which form a sort of inside out Reuleaux's triangle, as in this picture:

To convince yourself that the shape should be 'inside out', note that the arcs remain the same way up when scaled, but the triangle of midpoints is upside down with respect to the original triangle. Not a very clear explanation, I know, but drawing yourself a diagram should help.

Thankyou

sfwc
<><

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dan
Member since Aug-12-04
Aug-12-04, 08:04 PM (EST)
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3. "RE: Reuleaux's triangle"
In response to message #2
 
   yes, a diagram did help.
in pink is an equilateral triangle, around it in blue is Reuleaux's triangle and within (in red) is the shape described by sfwc.

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