Coordinates of object on circular path at an angle

Question

I think I have been looking at this too long... I need help finding the $(x,y,z)$ coordinates of an object on a circular path that is tilted at an angle of $30^\circ$ from the horizontal, in a similar manner to the example on the left.

Answer 1

This can be solved in Cartesian coordinates. Here I'm assuming that the $x$ axis directed rightwards and the $z$ axis is upwards, w.r.t your drawings.

The path is an intersection of a plane with the sphere, inclined at an angle $\theta$ to the $x$ axis. The plane is given by $$z = x \tan \theta.$$

The radius $r$ is $$r = \sqrt{x^2 + y^2 + z^2}$$ so $$y = \pm \sqrt{r^2 - x^2 - z^2}$$ $$y = \pm \sqrt{r^2 - x^2 - x\tan^2 \theta},$$ provided $r^2 - x^2 - x\tan^2 \theta \ge 0$.