I know how to find a point on a circle given a radius and an angle, but my knowledge of trigonometry doesn't extend much further than that. My question is probably best explained diagrammatically:
Given a centre point, radius, and two points on the circle, how do I find the 'middle' point between those two points? I'm also interested in knowing the distance between the internal line formed between the two points and this new middle point. (Referring to the diagram, I'm trying to determine the distance, q, and the unknown point, x?, y?)
Thanks!
Stewart
Best Answer
First, you have two points $(x_1,y_1)$ in your diagram. I will use that for the one on the circle. You can calculate $d$ from the coordinates of $(x_1,y_1)$ and $(x_2,y_2)$ as $d=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. If you draw the line segment from the center to $(x_1,y_1)$ you have a right triangle with sides $d, (r-q)$, and $r$, so $r^2=d^2+(r-q)^2$ which you can solve for $q$ by the quadratic equation. Then the coordinates of the intersection of the radius and the chord are $(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})$ and you can extend the line segment from the center through that until it has length $r$