# What is the proper geometrical name for a a rectangle with a semi-circle at each end?

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?

• This is a pretty cool question... I swear most of my favorite questions are from first time users. – BBischof Sep 26 '10 at 4:24
• See also recently graphicdesign.stackexchange.com/q/117005/129372 – rschwieb Nov 14 '18 at 14:33
• Also "pill/capsule" and my tongue-in-cheek suggestion ciiiiircle. – rschwieb Nov 14 '18 at 14:37

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.

• ooh, good hunting! – BBischof Sep 26 '10 at 4:24

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.

• Do track & field racetracks have the same properties I wonder? – Larsenal Sep 26 '10 at 4:28
• I 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. – J. M. is a poor mathematician Sep 26 '10 at 5:05
• @Larsenal, @J.M.: You can see from this satellite image that running tracks do have abrupt changes in curvature: maps.google.com/… – Tomer Vromen Sep 26 '10 at 9:02
• @Tomer: Are they banked or not? :) – J. M. is a poor mathematician Sep 26 '10 at 9:32
• I've never seen a track shaped like that. Here's Churchill Downs. I believe such tracks are banked for drainage. – Dennis Williamson Sep 26 '10 at 15:35
• Are you serious? – Rasmus Sep 26 '10 at 16:50
• I 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. – Jonas Meyer Jan 22 '11 at 2:43

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. :)

• While your observations about racing animals provide interesting color, a good Answer would involve at least a connection to mathematical reasoning or practice. – hardmath Sep 27 '18 at 3:48
• Each 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? – user597647 Sep 27 '18 at 13:32

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.