If you've seen a family tree you will have no doubt thought of something like this



This sort of object is called a graph. It is a set of points with lines connecting certain pairs.

Yet another type of parameterization is needed to account for, say, all possible positions of an object. Let's parameterize all possible positions of a bus in a square parking lot. Here we will want to account for distance, in that some positions are closer to others.



(In the first two situations the positions of the busses are "close." In the third they are not.)

We'll say two positions are close if they are close both in angle (i.e., direction the bus is pointing) and in location. How can we parameterize the set of all possible positions of the bus in the square parking lot? Let's first parameterize location. We need a point for each location, and we need to preserve information of distance. What does such a set look like?