What would be the steps or equation to graph a irregular convex equilateral polygon inscribed in a ellipse. I have not been able to find a example or equation of this online other then unanswered questions.
In my example we are trying to graph a convex equilateral Dodecagon. The ellipse is $$\frac{x^2}{20^2}+ \frac{y^2}{15^2} =1 $$
Is there a way to graph the Dodecagon? If so is there a equation to define this? Sorry if these are not the best tags
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Any regular polygon can be circumscribed Now let's assume it is possible to graph a regular dodecagon Then construct its circumcircle
Since the circumcircle and ellipse passes through each point it would imply the circle and ellipse have 10 intersections which is clearly absurd so by contradiction we cannot construct a inscribed regular dodecagon in an ellipse or any conic for that matter
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This is the equation that I found on desmos. This graphs the vertices of the polygon. I modified it to make it work as an ellipse instead of a circle by multiplying $a$ with $cos$ and $b$ with $sin$.
$$\left(\left(a\cos \left(2\pi \cdot \frac{\left[1...n\right]}{n}\right),b\sin \left(2\pi \cdot \frac{\left[1...n\right]}{n}\right)\right)\right)$$ This is the equation for the ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} =1 $$
$n=12\quad a=20\quad b=15$
This is the answer that got me what I needed. It is the Answer. Not the solution, I don't think this proves anything. The next step would be to measure the sides and see if its equal.
The choice of points given in your answer does not result in an equilateral dodecagon. Define a point on the ellipse as a function of the parameter $\theta$ $$f : [0,2\pi) \to \mathbb R^2, \\ f(\theta) = (20 \cos \theta, 15 \sin \theta).$$ Then your choice corresponds to an evenly spaced distribution on the interval $[0,2\pi)$, namely $$\theta \in \left\{ \frac{\pi k}{6} \right\}_{k = 0}^{11}.$$ But this is incorrect because the distance between $f(0)$ and $f(\pi/6)$ is not equal to the distance between $f(2\pi/6)$ and $f(\pi/2)$, which you can calculate explicitly: $$||f(\pi/6) - f(0)||^2 = \frac{3025}{4} - 400 \sqrt{3}, \\ ||f(\pi/2) - f(\pi/3)||^2 = \frac{1975}{4} - 225 \sqrt{3}. $$
The required spacing for a dodecagon with one vertex at $f(0) = (20, 0)$, can be found by numerically solving for $\theta, \psi$ in the system $$||f(\theta) - f(0)||^2 = ||f(\psi) - f(\theta)||^2 = ||f(\pi/2) - f(\psi)||^2$$ under the restriction $0 < \theta < \psi < \pi/2$, and the remaining vertices are simply reflections about the coordinate axes. Unfortunately, an elementary closed form solution for these parameters is not possible. Numerical methods yield (in radians) $$\theta \approx 0.59573630253877764269966203213567413136048611829430\ldots, \\ \psi \approx 1.1065501747316065924388786458228518171561303797842\ldots.$$ It is worth comparing these against $\pi/6$ and $\pi/3$. A plot of the resulting equilateral dodecagon is shown below in red, and the green points are the spacing that is evenly distributed with respect to $[0,2\pi)$:
Note also that this is not the only possible choice for an equilateral dodecagon inscribed in the aforementioned ellipse. Different choices for the "starting" vertex will result in different solutions, but the one illustrated here is the simplest in that it requires only two variables to solve. It is also worth noting that if the major or minor axes change, the whole system needs to be reevaluated, since the equilateral property is not invariant under a non-isometric scaling transformation.