Calculating the distance in degrees between two points on a circle

Question

In a video game I am I'm working on I'm trying to rotate an object around a secondary object. The secondary object will always be in the exact center and the rotating object will always rotate in a circular pattern. As the object rotates around the central object I need to determine how far along the circle the object has traveled (in degrees). Here is a more abstract representation of the problem:

Above I have two circles showing two examples of the problem I'm trying to solve. x1,y1 represents the starting point of the object that is rotating, x2,y2 represents the current position of the same rotating object and x3,y3 represents the object around which it is rotating. I need to calculate in degrees the distance traveled around the circle at any given point in the rotation. The object will never rotate around the circle more than 360 degrees. The radius will be variable but will be a known value that can be used in the equation. I will need an equation that can calculate the distance traveled up to 360 degrees.

Appreciate your help friends!

Answer 1

Note that the question is over-specified: If Point #1 and Point #2 and the radius are specified, the location of Point #3, the center, is already determined. So:

Firstly: Subtract $x_3$ from $x_1$ and $x_2$ and then $y_3$ from $y_1$ and $y_2$ This produces new co-ordinates $(X_1, Y_1)$ and $(X_2, Y_2)$

Check that $\sqrt{X_1^2+Y_1^2}=\sqrt{X_2^2+Y_2^2}=R$

where R is the radius of the circle (which is also a given quantity). If not, then you have bad data.

Let D the distance between the two points on the circle:$$D=\sqrt{(X_1-X_2)^2+{(Y_1-Y_2)^2}}$$

Then $\theta$, the angle between the radii going to the two points is given by $$\theta=2\times \sin^{-1}\left(\frac{D}{2R} \right)$$

This will give a value for $\theta$ between $0$ and $180$ degrees.

Answer 2

Let the radius (distance from centre two two corrdinates) be $r$.

Let the distance from $(x_1, y_1)$ to $(x_2,y_2)$ be represented by $c$.

$$c=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

If $x_1<x_2$ (from Law of Cosines):

$$\theta=\arccos\frac{2r^2-c^2}{2r^2}$$

If $x_1>x_2$:

$$\theta=360-\arccos\frac{2r^2-c^2}{2r^2}$$

Answer 3

Hint: the transformation from polar coordinates ($r$,$\theta$) to Cartesian ($x$,$y$) is given by: $$x=r\cos\theta$$ and $$y=r\sin\theta$$

where $r$ is the distance from the origin (in your case the unmoving object) and $\theta$ is the angle measured anti-clockwise from the positive $x$ axis.