In classical Euclidean geometry, the measure of the distance between two points, say A and B is calculated using the well known formula:
This is certainly the most intuitive notion of distance there is. However, it is not the only reasonable notion of distance. The theory of metric spaces is concerned with the differences and similarities of different definitions of distance. One interesting example of a redefinition of distance is the so-called 1st order Minkowski distance defining:
In simple terms, the distance is defined as the maximum of the vertical and horizontal distance of the two points. We might notice that this definition of distance is fairly natural: it is the minimum distance that a taxicab would need to travel to reach point B from point A, if all streets are only oriented vertically and horizontally. For this reason, this metric is often referred to as the Manhattan metric.
Many geometric figures, like circles, ellipses, parabolas, are defined
in terms of distance. For instance, a circle is just the set of points
whose distance to the center is the radius. These of course start to look
quite different if we use another measure of distance. The example of the
circle is probably the most famous one.
Another important geometric figure defined in terms of distance, is
the locus of points which are equidistant to two points A and B. In Euclidean
geometry, this is just the perpendicular bissector of the line segment
[AB]. The situation is not as simple in taxicab geometry. We need to consider
four basic examples:
![]() |
![]() |
|
|
![]() |
![]() |
|
|
Back to the main page |