All.
I've made a post similar to this before, but I have since come up with a different way to tackle the problem of creating a Geocentric model of the Solar System, this time using trigonometry to help us along.
I'm thinking that one could start with the current position of the Earth. For simplicity's sake of explaining the methodology, let's assume that Earth revolves in a perfectly circular orbit represented by $x^2+y^2=\alpha$. Let's take another arbitrary planet that also orbits in a perfectly circular orbit modelled by $x^2+y^2=\beta^2$. Based on this, we can assume that Earth, at some initial point, is at the positive $x$-axis, thus can be modelled at being at the point $(\alpha\cos0, \alpha\sin0)$ or $(\alpha, 0)$. The other planet would be astronomically unlikely to also lie exactly on the positive $x$-axis, so assuming that it starts at an angle of $\theta$ from there, we can place its point at $(\beta\cos\theta, \beta\sin\theta)$.
Now, if the Sun is at the origin, we can design a triangle with side lengths $\alpha$, $\beta$, and $\sqrt{(\alpha^2+\beta^2-2\alpha\beta\cos\theta}$. Since the more complicated side length represents that which is opposite the angle representing the angular difference, one can conclude that the angular difference is $\arccos\sqrt{\alpha^2+\beta^2-2\alpha\beta\cos\theta}$.
With this function of angular difference, (let's call it $A(a, b)$ with $a$ being the Earth's angle and $b$ being the other planet's angle), we need to define then a model that allows for $A$ to take functions of angular change of the planets. If we call these functions $p_n(t)$ where $n$ is the number of the planet progressing from the set of astronomical bodies one wants to consider and $t$ is simply time from the initial reading (both positive and negative), then we can find that $A(a+p_0(t), b+p_n(t))=\arccos\sqrt{\alpha^2 + \beta^2 - 2\alpha\beta\cos(a+p_0(t) - b+p_n(t))}$.
Where I'm stuck is in the development of a definition of $p_n(t)$ because it's individual to the planets' orbits. According to Kepler's Laws of Orbit, the amount of distance covered by a planet is equal in all intervals, however, this requires that planets move faster or slower based on their distance from the orbiting body. With that in mind, it is clear that there needs to be a factor of $\sqrt{\beta^2\cos^2\theta + \beta^2\sin^2\theta}$ in there (or $\beta$) that modifies the speed at which the planet orbits the body. However, this is where the simplification of the above-described model begins to fall apart because Kepler's Laws also state that orbits are elliptical with the body being orbited at one of the foci; the distance of $\beta$ changes over time. (This fact also stands for Earth's orbital distance of $\alpha$.)
Doing the same process, we can see that the function $A(a+p_0(t), b+p_n(t))$ has to be updated to include the general form of an ellipse, thus $$A(a+p_0(t), b+p_n(t))= \sqrt{\frac{\alpha^2(b_1^2\cos^2(a+p_0(t)) + a_1^2\sin^2(a+p_0(t)))}{a_1^2b_1^2} + \frac{\beta^2(b_2^2\cos^2(b+p_n(t)) + a_2^2\sin^2(b+p_n(t)))}{a_2^2b_2^2} - \frac{2\alpha\beta(b_1b_2\cos(a+p_0(t)) + a_1a_2\cos(b+p_n(t)))}{a_1a_2b_1b_2}}$$
and that aforementioned constant factor for the $p_n(t)$ function would have to be $$\beta\sqrt{\frac{\cos^2\theta}{a^2} + \frac{\sin^2\theta}{b^2}}$$
So, ultimately my question is how could one devise a function that makes the amount of the arc covered by the orbiting planet the same over any given interval of $t$ would be the same.
Attempt at the Question
Since the circumference of an ellipse is $4a\int_0^\frac{\pi}{2}\sqrt{1-(1-\frac{b^2}{a^2})\sin^2\theta} d\theta$. At this point, I'm lost. I would think that we need to track the rate of change of the planet's position as $\theta$ changes, but I don't know how to go about doing that. I think that reintroducing that factor of $\beta\sqrt{\frac{\cos^2\theta}{a^2}+\frac{\sin^2\theta}{b^2}}$, creating an equation of $$4a\beta\sqrt{\frac{\cos^2\theta}{a^2}+\frac{\sin^2\theta}{b^2}}\int_0^\frac{\pi}{2}\sqrt{1-(1-\frac{b^2}{a^2})\sin^2\theta}d\theta$$ I'm having trouble evaluating the definite integral to simplify that equation, and when I've looked it up it just refers to it as the 'incomplete elliptical integral of the second kind' and suggests the notation $E(\theta | 1-\frac{b^2}{a^2})$.
I've gotten this far, but I don't know how to progress from here. Thank you in advance for your help!