Computing an angle from x,y coordinates

Question

I'm trying to do collision testing between a finite line segment, and an arc segment. I have a collision test which does line segment vs. line segment, so I was going to approximate these arc segments with line segments and run my existing test.

The data I have defining the arc segment(s) are three points. Two of which are endpoints that lie on the circumference of a circle, and the third point is the center of that circle.

So far this is what I've got:

Let (a,b) be the center point of the circle, let 'r' be the radius of the circle, and (x1, y1), (x2, y2) be the endpoints of the arc segment which lies on the circumference of the circle.

The following parametric equations give the x, and y locations of an arc. 't' is the parametric variable.

x = a + r * cos(t) y = b + r * sin(t)

To create the line segments from the arc, I wanted to walk the arc for some fixed ratio of 't' creating line segments along the way, until I've reached the end of the arc. To do this I figured I'd have to find the start and end angle. I'd start walking the arc from the start angle, and end at the end angle. Since I know the start and end points I figured I could use these equations to solve for these angles. The following are my equations for this:

t = arccos(^(x-a)/r)

or

t = acrcsin(^(y-b)/r)

The problem I'm having is that the range of values returned by these functions (http://en.wikipedia.org/wiki/Inverse_trigonometric_function) is limited, so there is a high probability that the angle I'm looking for will not be returned because these functions are multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc

How do I get the exact angle(s) I'm looking for? Or, can you think of a better/different way of achieving my goal?

Answer 1

Take a look at the atan2 function, which should exist in whatever programming language or math library you're using. It takes two arguments, the x and y coordinates of a point (for you: (x-a)/r and (y-b)/r) and returns the angle in the range -π to +π.

Answer 2

Your pseudo-code looks a lot like Python. If you don't mind using Python I would recommend the Shapely Library. If you just want the algorithm, check the source.

Shapely objects have the 'simplify' and 'intersection' methods.

Answer 3

At least IMO, you're going at this the wrong way. A line has an equation y=mx+b. A circle has an equation x² + y² = r². You're looking for a point at which the x and y of the circle equals the x and y of the line. You can do that by substituting the mx+b equation for the line for the y equation in the circle, and then solve using the quadratic equation.

The equations involved do get a bit long, but quite a few web pages (e.g., http://www.sonoma.edu/users/w/wilsonst/papers/geometry/circles/default.html) have them, at which point it's simple matter of implementing the equations as a couple of functions and plugging in the values for your particular circle/line. A solution based on these equations complete avoids the ambiguity from using an arg tangent.

Answer 4

I actually ended up finding another answer:

http://stackoverflow.com/questions/1073336/circle-line-collision-detection

Hope that helps.