I am writing a game and need to figure out some math. If I have a circle with the equation $r^2 = (x-d)^2+(y-e)^2$, where $r$, $d$, and $e$ are constants, and a point $A(a,b)$, how do I find the point(s) on the circumference of the circle that are a given straight-line distance $D$ from $A$?
I’m in high school, so I don’t have the most extensive math knowledge. However, my first attempt was to find the $x$-values by isolating the $y$-value in the circle equation and substituting it into the distance formula. Then, I would simply substitute the $x$-values back into the circle equation to get the corresponding $y$-values. However, I can’t isolate $y$.
In the end, I will only use this where $A$ is on the circumference of the circle already.
If you’re looking for the “straight-line distance” $D$* from A to point(s) on the circumference, you would construct a second circle centered on point $A(a,b)$ with a radius of $D$, $D^2 = (x – a)^2 + (y – b)$. You would then set the equations for the circles equal to each other:
$$ (x – a)^2 + (y – b) – D^2 = (x – d)^2 + (y – e)^2 – r^2 .$$
*I’m changing the letter to $D$, since you are calling the x-coordinate of the center of A’s circle $d$.
This gives you information about the intersection points of the two circles. (One of the nice features of this is that the quadratic terms $x^2$ and $y^2$ will cancel out.) You will now have a relation between $x$ and $y$ that you can substitute into either circle equation to solve for one of the coordinates; the relation will then let you find the other. (There will be two, one, or no solutions, according to the number of intersection points.)