# Thread: Find a point on a circle

1. ## Find a point on a circle

Refer to the picture below, my problem/questions will follow

I have two points, one is the center of the circle and one is a point on the circumference of the circle.

What I need to do is find the position of points A, B, C and D given that the arc length between each points is 10

I'm trying to find the best method (least computational) way of doing this.

I have found a way that requires essentially 3 steps (which I will detail below) but I believe there is a better solution.

my solution:

Step 1 - Use the Arc Length equation to solve for the central angle C, we have the arc length (10) and we can get the radius r

Step 2 - Use the Arctangent function to obtain the angle of the known point on the circle's circumference in relation to the center point

Step 3 - Subtract the central angle found in Step 1 from the angle found in Step 2 and plot a point on the circle's circumference at this angle with a distance of the radius (also found in Step 1)

This works, but is a lot of math. I would think there is a more simple way to do this... does anyone have an idea?

2. You could setup an equation for the circle, and since you know the change in arc length every time, that means the change in degrees from the center is the same as well. If you know the degree change, you can use that in the equation of the circle to find the points.

3. Originally Posted by eXist
You could setup an equation for the circle, and since you know the change in arc length every time, that means the change in degrees from the center is the same as well. If you know the degree change, you can use that in the equation of the circle to find the points.
Sure, but the equation I will set up will be based on my findings so far. Which I don't believe to be the most efficient, what I'm trying to find out is if there are alternative ways to do this that may be better.

4. Well the radius of the circle is $\displaystyle 50\sqrt{2}$.

With that we can set up an equation involving $\displaystyle \theta$:

$\displaystyle x = 50\sqrt{2}cos\theta + 200$
$\displaystyle y = 50\sqrt{2}sin\theta + 200$

We know the point (150, 150) is located at 135 degrees or $\displaystyle \frac{3\pi}{4}$ (You can figure this out by setting x and y = 150 in the previous equations and solving for $\displaystyle \theta$).

Now that we have the starting position and the equation of the circle, we need to find out how many degrees/radians 10 arc length units equates to. We can do that by applying the arc length formula. For now, I'm going to call that $\displaystyle \theta_1$ (this would be the answer you get after applying the formula)

Once you figure that out, you can finally use your equation and displace the starting position by $\displaystyle \theta_1$ once, twice, three, and four times.

So for point A example:
$\displaystyle x = 50\sqrt{2}cos(\frac{3\pi}{4} - \theta_1) + 200$
$\displaystyle y = 50\sqrt{2}sin(\frac{3\pi}{4} - \theta_1) + 200$

That should give you the coordinates for point A