In a real-time environment such as an arcade game like Asteroids, it is imperative that
the world is processed as quickly as possible.  This means that the collision detection
algorithm must be fast.  The naive algorithm runs in O(p*q) time, where p is the number
of edges in polygon P and q is the number of edges in polygon Q.  In a world containing
several moving polygonal objects, there is a lot of time when there are no collisions, so
it is not always necessary to run a collision detection algorithm.  A method exists for testing
whether or not there is the possibility of a collision, in constant time.

The minimum spanning circle, or bounding circle, of a set of points is the smallest circle
which contains the set of points.  If we are testing for a collision between two moving
polygons, we can test to see if their bounding circles intersect, in constant time, before
running the more accurate, and costlier, collision detection.  If their bounding circles do
not intersect, there is no possibility that a collision exists.

Jon Rokne (1992) offers an algorithm to compute the bounding circle of a set of points
in O(n^2) time.

1.  Find p, the point with smallest y-coordinate, and q, the point which minimizes
     the angle between pq and the x-axis.

2.  Find r, the point which minimizes the angle prq.
 

3.  If the angle pqr ( or qpr ) is obtuse, swap p ( q ) with r and go back to 2.

4.  If the angle prq is obtuse, then the center of the bounding circle is described by the
     midpoint of pq, and the radius is half the length of pq.

5.  If all angles in triangle pqr are acute, then the circle is defined by the points p, q, r.

Collision detection can be further sped up by using a faster algorithm for computing
the existence of the intersection of two polygons.  Chazelle and Dobkin (1987) offer
an algorithm that computes the existence of this intersection in O(log n) time.
 

All operations described for the red polygon are also applied to the green polygon:

1.  Select a vertex p1 of the red polygon.  Select some point interior to the green polygon.
     Construct the line (yellow) which connects these points.  If p1 lies on the segment ab
     the intersection exists.

2.  If we haven't found an intersection yet, construct the black lines.  The upper black
     line is defined by p1 and the point on the portion of the green polygon above the
     yellow line which maximizes the angle formed at the intersection of the yellow and
     black lines.  The lower black line is defined in a similar fashion, using the points of
     the green polygon which are below the yellow line.  Repeating this procedure for
     the green polygon yields the following situation:

The problem has been reduced to testing the intersection of the red and green chains captured
within the trapezoid.