# Cartesian equation of a trochoid

A trochoid is defined by the following parametric equation:

$x = r\cdot\theta-d\cdot\sin(\theta)$

$y = r - d\cdot\cos(\theta)$

When $r = d$ the analytical form is

$x(y) = r\cdot\cos^{-1}(\frac{r-y}{r})-\sqrt{y\cdot(2\cdot r - y)}$

Is there also a cartesian form when $r <> d$?

The goal is to derive a formula for the intersection points of two arbitrary trochoids.

HINT:

You can more easily solve for $\theta_1, \theta_2$ from separate parametric forms without merging x- and y- equations the transcendental equations numerically

$x = r1\cdot\theta_1-d1\cdot\sin \theta_1 = r2 \cdot\theta_2-d_2\cdot\sin \theta_2$

$y = r1 - d1\cdot\cos \theta_1 = r2 - d2\cdot\cos (\theta_2 )$

• I do not quite understand: shouldn't it be: $x = r_1\cdot\theta_1-d_1\cdot\sin(\theta_1) = r_2\cdot\theta_2 - d_2\cdot\sin(\theta_2)$ and $y = r_1 - d_1 \cdot \cos(\theta_1) = r_2 - d_2\cdot \cos(\theta_2)$? But it is exactly the question how to find a solution for this.
– Ctx
Feb 19, 2016 at 22:59
• Indeed typo just about to correct it. Feb 19, 2016 at 23:07
• Thank you, but I still do not see a way how to solve these equations for specific $d_1,r_1,d_2,r_2$'s
– Ctx
Feb 19, 2016 at 23:19
• Some technique to handle self-intersecting muti-valued curves is to be found. Numerical/graphical method works. Feb 20, 2016 at 14:40