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I'm trying to derive a formula for a very particular egg curve. It should be taller on one side of the origin, but a line drawn from one side to the other passing through the origin should have a constant length regardless of the angle of the line.

For reference the data points below should help understand the shape.

0deg, x=0, y=10 45deg, hypotenuse=11 90, x=-12, y=0 135deg, hypotenuse =13 180deg, x=0, y=-14 270deg, x=12, y=0

As the data above shows, in this example the length any line drawn from one side to the other through the origin is always constant (24).

However I can't figure out how to make it work, I'm trying from sin\cos but they're always symmetric about 0... And an offset won't work as it needs to be 0 at 90 degrees.

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    I don't know if the figure you need does exist: I'm finding only cardioid-like curves. – 2017-01-13
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    Just to make sure I understand your figure, is it correct that your "0deg" is what convention would normally call $90$ degrees and that the radius of the figure increases linearly from $10$ to $14$ as the angle goes from $0$ to $180$ degrees? – 2017-01-14
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    Writing the curve in polar form, it seems that the desired condition is that $r(\theta)+r(\theta+\pi)$ be independent of $\theta$. But we also need $r(\theta)>0$ for all $\theta$ and (more importantly) $r(\theta+2\pi)=r(\theta)$ for all $\theta$. There is one obvious solution $r(\theta)=$const., corresponding to a circle at the origin; the question is what others exist. – 2017-01-14
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    Following that idea, it looks like polar curves of the form $r(\theta)=a-b\cos\theta$ should work if $a>2b>0$. (If $0$\theta=0$ and therefore look more like a cardiod than an egg; this is consistent with @N74's remark above.) – 2017-01-14
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    This came up as a suggestion on my own question. I found $F\left(\theta\right)=\frac{\left|\theta-\pi\right|}{\pi}$ gives a palm-leaf shape with constant width through the origin. It's as close as I can get to your problem. – 2018-05-21

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