Similarly to the article about intersection points of two circles we're now interested in the area that is formed by the intersection of two overlapping circles. We could formulate cases to step through the same as in the other article, but I will do it a little shorter this time.

How can you find the intersection points of two circles, given by the two center points and their radii? More formally: Given two circles with their center points \(\vec{A}\) and \(\vec{B}\) and their radii \(A_r\) and \(B_r\), we want to find the points \(\vec{P}_1\) and \(\vec{P}_2\) which represent the intersection points of both circles. Obviously, there can be one, two, infinity or no intersections at all, namely when one circle is completely within the other or the areas of the circles does not overlap. One way would be to formulate the two circles as their circle-formulas, subtract them and work out the x and y values. Quite simple, but needs a lot of calculations if you want to keep it fully parameterizable. Here is a different way:

There was a time when tag-clouds were the thing for website owners to fancy oneself. These clouds are mostly gone, but seen from the perspective of how to implement such a thing, one can learn quite a lot, especially with large amounts of links. Anyway, imagine you publish some articles on your website, which are stored in a table "post" and you want to to add tags to every post in order to print a tag-cloud.

Imagine, we store bit-flags in a database as integers and want to update the bit-set by simultaneously deleting and updating the bits. An expression like this can quickly become unhandy. However, I wanted to make this solution pretty and after some tinkering I came up with a quite nice solution.

GPS is probably the best known global positioning system, even if there are other countries besides America who built up their own, like Europe with Galileo, Russia with GLONASS and the newest, BeiDou in China. All are based on the same technique, a triangulation of spheres, whose radius is the signal transfer time, given by an atomic clock on the satellite. In general, the more satellites you use this way, the better the accuracy of the determination of the location and some receivers already combine the varying systems.

Imagine, you have a coordinate system with some points. Parallel to the x-axis is a line, which is connected to each of the points. Your goal is to figure out what the minimum accumulated distance of the points to the line is by moving the horizontal line up and down. Look at this illustration and use the mouse to get a feeling for it:

Complex numbers sound complex, but in fact they are the most natural numbers - even if the name is a bit misleading. When extending the natural numbers to integers, to rational numbers, to real numbers and finally to complex numbers; it's the first field in which every operation is well defined.