First I am a newbie in maths so please forgive me if I am not as rigorous as you would like, but do not hesitate to correct me.
I want to find the equation of a torus (I mean the process, not just the final equation that I can find on Google). Knowing that a torus is the set of point on the circles having all their centers on another circle I came with something like:
Let $C_c$ be the "central" circle with a radius $R$ and and center $P_c(a, b, c)$. Also, let $M_1(x_1, y_1, z_1)$ be all the points on $C_c$. Let $C_a$ be an "auxiliary" circle (one that has $M_1$ as a center), $r$ his radius and $M_2(x_2, y_2, z_2)$ a point on that circle.
I'm looking for all the points $M_2$ to find the torus. Here's what I came to:
\begin{cases} (x_1 - a)^2 + (y_1 - b)^2 - R^2 = 0 \text{ (1)}\\ (x_2 - x_1)^2 + (y_2 - y_1)^2 - r² = 0 \text{ (2)} \\ \end{cases}
And I am stuck here, how can I transform those equations into a parametric form or a cartesian equation?
Thanks.
EDIT :
My goal is to find $x_2$ and $y_2$ here. So I decided to calculate $x_1$ and $y_1$ to use them in $(2)$.
From $(1)$ I get something like $x_1(x_1 - 2a) = -a^2 - y_1^2 - b^2 + 2by_1 + R^2$
But I am stuck here since I don't know how to "isolate" $x_1$