Calculate point on circle perimeter with just radius, center point and X or Y offset? How to calculate point on circle perimeter that is Y (or X) offset from another point on the perimeter? The center point, radius and offset are known.
Sorry i have had no success googling this, maybe i am just using the wrong terminology.
Circle problem picture
 A: Going off the picture, I am going to assume we are given a center point, which I will call $C=(x_c,y_c)$, the radius, $R$, and a point $(x_0,y_0)$ on a circle. The challenge being to find a point on the circle that has a $y$ coordinate of $y_0-Z$ (i.e. is $Z$ units below the given point).
Well using the given information we have the equation
$$(x-x_c)^2+(y-y_c)^2=R^2\qquad (1)$$
As an equation for our circle. We know the $y$ coordinate of the point we're trying to find, which is $y_0-Z$, and so we need to use $(1)$ to find the $x$ coordinate. Plugging in this $y$ coordinate we have
$$(x-x_c)^2+(y_0-Z-y_c)^2=R^2$$
Solving for $x$ we get
$$x=x_c\pm\sqrt{R^2-(y_0-Z-y_c)^2}$$
So our coordinate is
$$\text{Desired Coordinate}=\left(x_c\pm\sqrt{R^2-(y_0-Z-y_c)^2}\,\,,\,\,y_0-Z\right)$$
Distinguishing between the $+$ or $-$ just depends if your want you point to be to the right (choose $+$) or left (choose $-$) of the center of the circle. In the picture it was to the left so we choose $-$.

If you are given an $x$ offset instead of a $y$ offset do this exact same procedure and you should get something that looks the exact same but with the roles of $x$ and $y$ switched.
