Circle-circle intersection

Intersecting circles can be done in a number ways. One approach is algebraic: take the formulae for the circles and solve them to obtain values for the x and y coordinates. Another approach derives the line that joins the points of intersection then intersects that line with the circle.

But the method I use is more direct, and I think mathematically more robust. By direct I mean it calculates the result directly and does not depend on intermediate results, such as calculating the line or one of x and y first. As such there's much less chance the order of calculation will affect the result.

Perp dot product

I have used the perp dot product in a couple of posts but not explained it. It has a page at Mathworld but no Wikipedia article so I should perhaps explain it here.



The perp dot product is a product between two vectors in two-dimensions, and is obtained by taking the dot product of one vector with the perpendicular of the other. The perpendicular is simply a vector at right angles to the vector it is based on with the same magnitude. It is obtained by rotating through 90° or π/2 radians, or by multiplying by the complex number i in the complex plane.

Lines

One problem that's been touched on in previous posts is how to deal with straight lines. In particular how to represent them efficiently and accurately. Making the wrong choice is not that disastrous but it can cause problems in some situations.

Angle between two vectors

Another use for inverse trigonometric functions is finding the angle between two vectors. Again there is more than one way to do it, and again the most straightforward approach is often not the best.

Inverse trigonometry

The functions asin, acos and atan are together the three principal inverse trigonometric functions; that is they are the inverse of the sine, cosine and tangent functions, the commonest trigonometric functions that describe all the ratios of the sides of a right-angled triangle.

New screenshot

A screenshot from something I'm working on. Yes, it has circles in it.

And a tip, if you're developing on a Mac. Turn on Web Sharing (under Sharing in System Preferences), then set the path Flash builds the SWF and HTML to something like "../../Sites/xxx", where xxx is the file name of each. Then you can access it in the browser using the IP address of your machine and the HTML file name, as above. No need to upload it to a separate site for network testing. It's still a local file so will work locally in the debugger or standalone Flash player just fine.

Circle-line intersection

Yesterday's method for determining whether a circle crosses a line can also be used to find where the crossing points are. The approach uses the distance calculated in the hitLine function with some simple circle geometry.

Circle-line incidence

Problems involving circles and lines can be straightforward, if handled the right way. The main reason for this is the fact that the distance from a circle to a line is just the distance from its centre to the line minus the radius. I.e. if a circle radius 100 touches a line the circle's centre must be 100 units from the line.

More on circles

Yesterday I described circle-point and circle-circle intersection tests. These are very straightforward to do, and it is interesting to look at why this is so, as it can help solve other problems with circles.

Circles

Circles are used for many things in 2D games. Not only for things that are inherently circular (wheels, balls) but for many effects and to describe many situations. For example the range of a gun is often a circle, or at least circular.