Polar radius and position vector: two-dimensional kinematics for high school students

The position vector $boldsymbol{r}(t)$ is the parametrization of the curve it rides on. But converting this to a polar form $rho(theta)$ is also a parameterization of the same curve.

For example an ellipse can be parameteized with

$$ boldsymbol{r}(t) = pmatrix{x(t) y(t)} = pmatrix{ a cos t b sin t} tag{1}$$

This position vector objeys the equation of the ellipse $$ left( tfrac{x}{a} right)^2 + left( tfrac{y}{b} right)^2 = 1$$ where $a$ is the semi-major axis, and $b$ the semi-minor axis.

Now consider the polar coordinates

$$ boldsymbol{r}(t) = pmatrix{x(t) y(t)} = pmatrix{ rho cos theta rho sin theta} $$

that yields the solution

$$ rho(theta) = frac{ a b}{sqrt{ a^2 - (a^2-b^2) cos^2 theta}} tag{2}$$

Expressions (1) and (2) are equivalent to each other since both describe the same ellipse.