Wikipedia:Reference desk/Archives/Mathematics/2016 July 1

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July 1

Egg-shaped curve

By tinkering around with Grapher, I hit upon a simple formula for an oval with an axis of length a whose shape (if a = 1) is very close to that of a chicken egg: ${\sqrt {x^{3}}}-{\sqrt {a}}x+y^{2}=0$, or ${\sqrt {x^{3}}}-{\sqrt {a}}x+y^{2}+z^{2}=0$ for an egg-shaped three-dimensional ovoid. Is this a function well-known for its shape? Does it have a name? I thought a formula for this shape had once been mentioned in Egg as food – do I misremember? Maybe I confuse it with Heart symbol, which does mention formulae for the shape. Oval does show an egg-shaped curve, but it's not as close. --Florian Blaschke (talk) 03:05, 1 July 2016 (UTC)

There are a few named egg-shaped curves, for example [1]. The same site lists many curves of different shapes, so you might want to check the index for more (under O for oeuf or oval). --RDBury (talk) 07:07, 1 July 2016 (UTC)

Thanks. Granville's egg is extremely similar to mine, but uses a completely different formula AFAICS. My formula gives a possibly even better shape with a = 1.5 or 2, by the way, but real chicken eggs seem to differ among themselves in their proportions too: some are thicker, more like the shape with a = 1. However, in a real egg, the big end seems to be slightly rounder, almost exactly like a sphere segment, while my curve is slightly flatter at its big end. Also, the little end of a real egg may by slightly pointier. However Granville's egg seems to share the same flaws; I don't think it is a significantly better approximation of the shape, but still the best I've seen besides mine. I've managed to get a possibly even better shape with a Cartesian oval, though, by fiddling with the tool on here; too bad it didn't show the values to plug into the formula. --Florian Blaschke (talk) 01:32, 2 July 2016 (UTC)

The curves studied by mathematicians tend to those with particular geometric, or sometimes algebraic, properties. It might happen that such a curve resembles an egg so it's named a egg or oval, but the point wasn't to find a curve shaped like an egg but to study the geometric property. (The word 'oval' comes from the Latin ovum so oval and egg-shaped are actually the same thing.) Similarly the bell curve arises from the study of probability, not from an attempt to describe the shape of a bell. The actual egg shape is the product of evolution, so it approximates an optimal shape according to what helps chickens survive; what factors might be involved is more a question of biology than mathematics. Note though that a bird egg is only one type of egg, and eggs in general range from spherical, to ellipsoidal, to the odd, pillow-shaped contraptions produced by sharks. The sphere minimizes the surface area for a given volume, and presumably one factor influencing egg shape is to not waste whatever material the shell is made of, but since all eggs aren't spherical there seem to be other factors involved as well. --RDBury (talk) 11:22, 2 July 2016 (UTC)

I do realise all that. I did not derive that formula theoretically, I just played around with variations on the circle $x^{2}+y^{2}=r$ and sphere $x^{2}+y^{2}+z^{2}=r$ formulae trying to create interesting geometric shapes. And I just happen to be fascinated by the specific shape of the chicken egg, because it is very peculiar and not a standard geometric shape, so I wonder if there's a formula, or construction, to create a curve that approximates it as exactly as possible. This may be much more interesting from a biological than mathematical point of view of course. I read it's actually simply the shape of a hen's uterus when it's stretched as far as possible (safely). I realise the problem is not very much of interest for mathematicians except as part of recreational mathematics. --Florian Blaschke (talk) 16:49, 2 July 2016 (UTC)

Isogonal conjugates Collinear with Incenter

Can two isogonal conjugates ever be collinear with the incenter in a non-isosceles triangle ? (The trivial/degenerate triplet incenter—incenter—incenter is also excluded). — 86.125.208.88 (talk) 06:51, 1 July 2016 (UTC)

I think this occurs for any triangle for any point on one of the angle bisectors. By the isogonal conjugate article, in trilinear coordinates the conjugate of x:y:z is 1/x: 1/y: 1/z. By the trilinear coordinates article, the incenter is at 1:1:1, and three points are collinear if and only if the determinant of the matrix whose rows are the respective coordinates is zero:

$D={\begin{vmatrix}1&1&1\\x&y&z\\1/x&1/y&1/z\end{vmatrix}}$ = 0.

After I work this out I get (z–y)x² – (z–y)(z+y)x + yz(z–y) = 0. Now z=y (the bisector of angle X) satisfies this. For the case of z≠y, solving for x gives the two solutions x=y (the bisector of angle Z) and x=z (the bisector of angle Y). Loraof (talk) 15:58, 1 July 2016 (UTC)

Resolved