first time poster here. I'm working on a math problem for a software application I'm writing. The problem is as follows:
You are given a circle with center point O and radius r. There are three points on the circumference of the circle. The points are A, B, and C. You know the x any y coordinates of each point, as well as the point's polar angle (0 to 2pi). Using these points, a circular arc is drawn along the circumference of the circle from A to B to C. Remember, the arc is circular so you cannot turn back at point B and head the opposite way to C. It is possible for the path to pass through the 0 degree point, and the path may have a CW or CCW direction, which is known. I need to use this information to determine if the arc has arc length greater than, less than, or equal to pi*r (so: is the arc longer or shorter than half a circle).
Here are a few other details [switching to degree mode here]:
-Point B is usually never further than +/- 20 degrees from point A.
-The angle difference between B and C can be more than 180 degrees so simply using arccos() calculations will not be suitable for this problem because the output of arcsos() has 180 degree extremes.
-The solution most not be iterative because there are hundreds of other complex calculations happening before and after this one, especially in the animations.
Thanks for the help
- P.S.
- Through some calculations irrelevant to this problem, I was able to determine the angle difference between A and B by finding a 'dTheta' value and ADDING it to the polar angle of made by point A. So if angle A is 10 degrees, and dTheta is -15 degrees, angle B would be 10 + (-15) = -5 degrees, but in my polar system, this can easily be converted to 355 degrees. Similarly, if we know dTheta is negative, we can keep subtracting [or adding negative] degrees from A or B and end up (while still following the arc's trajectory) at angle C (or some angle that is C +/-360).
Best Answer
Here are some thoughts. Suppose that you have two points $A$ and $B$ which lie on a given circle of radius $r$. If $A$ and $B$ are diametrically opposed, then one can chose either half-circle (relative to the diameter $AB$) to get from $A$ to $B$, and the length of this path is $r\pi$. In general, with the aid of a picture, the arc length from $A$ to $B$ is $r\lvert\theta_B-\theta_A\rvert$, where $\theta_A$ and $\theta_B$ are the angles of $A$ and $B$, respectively. This is fairly easy to see by translating and scaling so your circle is centred at the origin and has radius $1$.
If one introduces a third point, $C$, then the total path length will be $r\lvert\theta_C-\theta_B\rvert + r\lvert\theta_B-\theta_A\rvert$, assuming, of course, that one traverses from $A$ to $B$, and then $B$ to $C$ (regardless of any change of direction at $B$.) If $A$, $B$ and $C$ satisfy $\theta_A<\theta_B<\theta_C$, then the arc length from $A$ to $B$ to $C$ is precisely $r(\theta_C-\theta_A)$. One must be careful if your arc "wraps over" the positive $x$-axis. (Adding another point at $(1,0)$ and the absolute angle above will help with that case.)
Note that your angles must be in radians for the above to work.
Edit: A bit more thinking about the problem yields the following. Rotate everything so that $A$ is at $(1,0)$ (so $\theta_A$ is now $0$, $\theta_B$ is now $\theta_B-\theta_A$ ($\pm2\pi$ if necessary) , and $\theta_C$ is now $\theta_C-\theta_A$ ($\pm2\pi$ if necessary)). Then $A$ and $C$ divide the circle into two arcs, say $C_1$ and $C_2$, and, unless $C$ is at $(-1,0)$, the arc lengths of $C_1$ and $C_2$ are different. Now $B$ lies on exactly one of these arcs, which we may assume is $C_1$.
Now the geometry is done, we just need to figure out the length of the longer arc and the shorter arc. This is easy since the shorter arc is $r\theta_C$ if $0<\theta_C\leq \pi$, and $(2\pi - \theta_C)r$ if $\pi<\theta_C<2\pi$. (The other arc has length $2\pi r$ minus the shorter arc length.) This should be relatively simple to code.