The most general equation of a straight line is
$$\vec{r}=\vec{r}_0+\vec{a} s$$
where $s$ is a parameter, the line passes through point $\vec{r}_0=(x_0,y_0,z_0)$ and is parallel to vector $\vec{a}=(a_x,a_y,a_z)$. In Cartesian coordinates, the parametric equations are
$$x(s) = x_0+a_x s$$
$$y(s) = y_0+a_y s$$
$$z(s) = z_0+a_z s$$
Write the parametric equations of this straight line in cylindrical and spherical coordinates, i.e. $r(s) = …$, $\theta(s) = …$, $z(s) = …$
and in spherical coords. $r(s) = …$, $\theta(s) = …$, $\phi(s) = …$
I tried using the conversion formulae (e.g. $\theta=\tan^-1(y/x)$ etc.) but arrived at long expressions. Is there a simpler way?
Best Answer
In cylindrical coordinates, from
$$r\cos\theta=r_0\cos\theta_0+sr_a\cos\theta_a,\\ r\sin\theta=r_0\sin\theta_0+sr_a\sin\theta_a$$
you draw $$r^2=r_0^2+2sr_0r_a\cos(\theta_0-\theta_a)+s^2r_a^2,\\ \tan\theta=\frac{r_0\sin\theta_0+sr_a\sin\theta_a}{r_0\cos\theta_0+sr_a\cos\theta_a}.$$
This does not simplify further.