I have to find the necessary translations in X and Y to move a point 0n a circle to another one.
I have a center (X and Y coordinates), a radius, and a current position in radians. And given a value in radians (the amount I want to translate), I have to find the amount of values in X and Y I have to move to get to that position.
So, for example, if default values are (2, 3) for the center, the radius is 3 and the starting radian position is 0. The starting point will be (5, 3) and I if I want to move the position 0.2 radians, I want to know how many units of X and Y I have to make to go to that position.
This is a simple drawing of what I want to do. Also, check out which one is the 0 radian starting position (rightmost part)
Best Answer
Let us first work with the example you gave and generalize from there.
You have a circle with center $(2,3)$ and radius $r=3$. You want to rotate the point $(5,3)$ on the circle by $\theta=0.2$ radians. To do this we parametrize the circle as $(x,y)=(2+3\cos\theta,3+3\sin\theta)$. The point $(5,3)$ has $\theta=0$ and we want to increase that angle by $0.2$. Thus the new point is $(2+3\cos(0.2),3+3\sin(0.2))\approx(4.9,3.6)$.
Now, in the general case, say you have a circle with center $(a,b)$ and radius $r$. The position of the initial point is $\theta$ radians along the circle from $(a+r,b)$. The parametric equation for the circle is $(x,y)=(a+r\cos\theta,b+r\sin\theta)$. Say you want to increase by $\phi$ radians. Then the new point is $$(a+r\cos(\theta+\phi),b+r\sin(\theta+\phi))$$