Q:

Find a point on a circle given a point and height

Point on a circle

Given :
A point on a circle (point and angle), 
radius of the circle ,
height (Orthogonal to horizon )
I would like to find
A point on a circle
or , and
The angle between the given point (x,y) and the new point (?,?)
Thanks !

A:

This is just a wild guess, given that that intermediary point is just $(x, y-h)$. 
Given $\alpha$ and $r$, calculate $(x,y) = (r \cos(\alpha), r\sin(\alpha))$ (plus an offest $\vec{p_0}$ if the circle is not at $(0,0)$.
Then, subtract $h$ from the y-coordinate, which should give you the point $(x,y') = (x, y-h)$:

If the new $y'$ coordinate is fixed then calculate the correct $x'$ coordinate by solving the circle equation in cartesian coordinates
$$x'^2 + y'^2 = r^2$$ 
with $y', r$ fixed and take the $x'$ that has the same sign as $x$. 
You will receive $(?,?)$ = $(x',y')$.
If you have the new point you can calculate the angle between the points $(x,y)$ and $(x', y')$ as
$$\alpha' = \text{atan2}\left(y'-y, x'-x\right)$$

