I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name? Does it begin with e?
What is the proper geometrical name for a a rectangle with a semi-circle at each end?
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2This is a pretty cool question... I swear most of my favorite questions are from first time users. – 2010-09-26
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0See also recently https://graphicdesign.stackexchange.com/q/117005/129372 – 2018-11-14
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0Also "pill/capsule" and my tongue-in-cheek suggestion *ciiiiircle*. – 2018-11-14
5 Answers
Obround, apparently. I don't know Wiktionary's source. This definition of obround does not appear in OED, for example. Googling indicates that this definition is commonly used for machine parts having this shape.
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0ooh, good hunting! – 2010-09-26
I'm pretty sure they actually use four clothoid arcs joined together in practice, e.g. this. This has a lot to do with the fact that the clothoid is the curve whose curvature is directly proportional to its arclength; an abrupt variation in curvature would equate to an abrupt variation in centripetal force, which can be bad for the racehorses (or even racecars, for that matter).
Here's a simulated clothoid track drawn in Mathematica:
Just to show that the bends are honest-to-goodness clothoids, I drew the clothoid corresponding to the lower right portion of the track in full (the dashed gray one).
The parametrization used is
$$(x\qquad y)=\left(\sqrt{\frac{\pi}{2}}C\left(\sqrt{\frac{2}{\pi}}s\right)\qquad \sqrt{\frac{\pi}{2}}S\left(\sqrt{\frac{2}{\pi}}s\right)\right)$$
where $C(x)$ and $S(x)$ are the Fresnel integrals; I leave you to verify using those expressions that the curvature of the clothoid is indeed directly proportional to the arclength.
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0Do track & field racetracks have the same properties I wonder? – 2010-09-26
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0I suppose so; for something going really fast around a track (whether that something be a car, a horse, or a sprinter), you'd want the property of the curvature not varying abruptly. I'm told even road bends (where the cars don't go *that* fast) are constructed to be (approximately) clothoidal so that the turns aren't very jarring. – 2010-09-26
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0@Larsenal, @J.M.: You can see from this satellite image that running tracks _do_ have abrupt changes in curvature: http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=39%C2%B059%E2%80%B230%E2%80%B3N+116%C2%B023%E2%80%B226%E2%80%B3E&sll=47.767945,-123.611298&sspn=0.687657,1.778412&ie=UTF8&ll=39.982254,116.393285&spn=0.003062,0.006947&t=h&z=18 – 2010-09-26
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0@Tomer: Are they banked or not? :) – 2010-09-26
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0I've never seen a track shaped like that. Here's [Churchill Downs](http://maps.google.com/maps?f=q&source=s_q&hl=en&q=Churchill+Downs,+Louisville,+Jefferson,+Kentucky+40215&sll=29.728835,-99.761448&sspn=0.050087,0.078106&ie=UTF8&cd=3&geocode=FRzsRgId8z7j-g&split=0&hq=&hnear=Churchill+Downs,+Louisville,+Jefferson,+Kentucky+40214&view=map&ll=38.203048,-85.769856&spn=0.005202,0.009645&t=h&z=17). I believe such tracks are banked for drainage. – 2010-09-26
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0After more digging around: apparently for the "cheaper" :P tracks, they use the clothoid sections only for joining straight (zero curvature) and circular (nonzero constant curvature) sections of track. I guess it depends on what the engineers were thinking when they were designing the tracks. – 2010-09-26
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0So maybe not always something entirely composed of clothoid arcs, but track/road designers do use it at turns. (Just to be clear on the picture: the black outline is *the* clothoidal track; I only drew the full clothoid in gray for illustration, but no designer in his right mind would use the spiraling portion of the curve. ;P ) – 2010-09-26
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0Roads and railways use clothoids. Civil engineers even have a thing called a "railroad curve". Running tracks just use two straight lines and two semi-circles (otherwise the hundred meters would be run on a curve). Maybe times for 200m and up would be faster if they used clothoids. Cycle tracks are usually very heavily banked. I don't know anything about horse tracks. – 2013-01-05
It's called a stadium. See http://en.wikipedia.org/wiki/Glossary_of_shapes_with_metaphorical_names or http://mathworld.wolfram.com/Stadium.html
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1Are you serious? – 2010-09-26
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1I don't know why John Bentin never responded to Rasmus, but it is apparently called a stadium by some mathematicians. I don't have a definitive source, but it was called a stadium by Kannan Soundararajan in his invited address at the Joint Mathematics Meetings in New Orleans earlier this month. – 2011-01-22
An oval. Racetracks, arenas, stadiums and round pens have one things in common - the shape, and the animals who use them at different speeds. The spaces are created based upon the strides of an animal creating velocity and balance as they move through their gaits of walk, trot, lope/canter to gallop. These forces can be seen in action during any training such as round pen training or barrel racing where the rider uses the shape, spacing and pattern angles to achieve speed and then cuts back, creates a circle and curves in speed to the next barrel. I believe it is called an oval, but then again I am just a cowboy. :)
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0While your observations about racing animals provide interesting color, a good Answer would involve at least a connection to mathematical reasoning or practice. – 2018-09-27
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0Each stride an animal takes is a certain distance covered at a certain speed, F/P/S similar to M/P/H on a straightaway, circle or oval shape. To me, riding horses in a line, in a circle or an oval or even a right angle brings math to my mind each time. Leaning, moving, shaping the pattern in and balanced enough for me to lope this corner, make this jump, overtake this cow? – 2018-09-27
Another shape called " Bermuda Bottle" has meridian curvature proportional to $x$. Also ratio of shell curvatures is $2$ when rotated about minor axis, however minor axis measures only $\approx 0.6 $ times major axis.
EDIT1:
It is is same shape as filled parachutes.
The differential equation of Cornu Spiral by J.M. is not a mathematician is ( $s$ is arc ):
$$ \frac{d\phi}{ds} = s/a $$
and the Bermuda Bottle (not sketched) is
$$\frac{d\phi}{ds} = x/b $$
They are similar in appearance. Clothoid curve has a third order discontinuity at sharpest corners.