Distance is a common concept that we use day to day. It is a quantity that represents the length between two points. The most typical image of distance is that it is the length of a straight line drawn from one point to another.


However, if you think about it carefully, even in everyday life, the word "distance" often refers to something other than a straight line. For example, when we talk about the distance from home to a train station, we usually think of the length of a walk along streets. When we say that a place is "far away," we are often thinking of the time it takes to travel there, not just the spatial distance. Kohei Sugiura's "Time Map", in which the map is deformed based on the time required to travel from the origin, visually expresses this everyday feeling. It is one of the most common methods of infographics to depict human relationships, similarities of data, etc., by replacing them with spatial distances.



The flight distance from Tokyo to Sao Paulo is not a straight line distance between two points, but a curved length along the surface of the earth. This is called spherical geometry, and the shortest distance between two points is the length of an arc on the intersection of the sphere and a plane passing through the center of the sphere (the great circle). In Einstein's world of relativity, the mass of matter bends space, so the shortest line between two points is also drawn along this curved space.



This way, distances can be measured in different ways depending on context and purpose. In mathematics, a metric space is a combination of a set and a distance function that determines the distance between the elements of the set. Technicalities aside, there are many interesting graphic possibilities when considering various functions for defining distances.

このように距離は文脈や目的によって違う測り方をすることができます。数学では集合に対してその元の間の距離を定める距離関数(distance function)を組み合わせたものを距離空間(metric space)と呼びますが、厳密な話はさておいてこの距離を求める関数を色々考えてみると図形的にも色々面白いことが考えられます。

The Euclidean distance


The Euclidean distance is the most intuitive, ordinary linear distance between two points.