Section 3.3 Using Taxicab Circles
¶Taxicab circles can be used to determine the distance between objects. For example, what is the distance \(d_T(P,l)\) from a point \(P\) to a line \(l\text{?}\) Distance in this context means the length of the shortest path connecting \(P\) and \(l\text{.}\) This distance can therefore be determined by drawing a small taxicab circle around \(P\text{,}\) as shown in Figure 3.3.1, then expanding the circle by increasing its radius until it touches \(l\text{.}\) Try it! What differences do you notice depending on the slope of the line?
Taxicab circles can also be used to determine the set of points that are equidistant from a given object, that is, the “circle” whose “center” is the given object.
For example, to determine the locus of points that are 6 blocks from the line \(l\) shown in Figure 3.3.2, first draw a taxicab circle around any point on the line, as shown. Then slide the circle along the line. Try it!
These notions of sliding and expanding circles are very useful when constructing objects satisfying a wide variety of conditions determined by distance.