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I recently discovered 'egg' curves, such as the following

$$\frac{x^2}{a} + \frac{y^2}{b}t_i(x) = 1$$

where $t_i$ could manifest as $t_1(x) = 1 + cx$ or $t_2(x) = e^{cx}$, with $c \geq 0$.

I was wondering if a parametric form exists, similar to the ordinary ellipse ($x = a\cos(t)$ and $y=\sin(t)$ for $t [0,2\pi]$), such that the shape can be plotted depending on the angle, rather than the range of x and y.

Example of my current approach (for $t_1(x)$):
$$bx^2 + ay^2(1 + cx) = ab \Rightarrow y^2(a + acx) = ab - bx^2 \Rightarrow y = \pm\sqrt{\frac{ab - bx^2}{a + acx}}$$

Example plot showing then Egg curve based on the above equations, with $t_1 = 1 + 0.2$

Тyma Gaidash
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Thomas
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  • Could you please give an explicit formula for $t_i(x)$? Try solving $$y=x\tan\theta, f(x,y)=1$$ for x and y in terms of $\theta$ – Тyma Gaidash Sep 26 '21 at 21:32
  • Thanks for the comment. Let's assume that $t_i(x) = 1 + cx$. Solving $\frac{x^2}{a} + \frac{(x\tan\theta)^2}{b}(1 + cx) = 1$ results in solving for cubic roots. – Thomas Sep 27 '21 at 09:20
  • See here for more information. This gets you an angle theta line and its intersection with $f(x,y)=1$ with $x(\theta),y(\theta)$ as the distance from the intersection to the $x$ or $y$ axis like the trig functions. I hope this works, tell me if not. – Тyma Gaidash Sep 27 '21 at 18:24

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