Geometry

# How do you determine distance in geometry?

The distance depends on the nature of the space (flat plane, the surface of a sphere), as well as the metric that is defined on it.

The distance between two points on the surface of the earth - unless they are very close to each other - is not be the straight line joining them since that would need a tunnel through part of the planet! Instead it is along the arc of the great circle. The great circle is the circle whose centre is at the centre of the earth and which goes through the two points.

For a 2 or 3-dimensional space, there are many different metrics possible. The traditional metric is the Pythagorean distance. In 2-dimensions, the Pythagorean distance between A = (x1, y1) and B = (x2, y2) is

sqrt[(x1-x2)^2 + (y1-y2)^2] with the 3-d counterpart analogously defined to include a (z1-z2)^2 term.

However, in 2-d, one popular metric is the Minkovski or Taxicab metric. Inspired by the grid-like layout of Manhattan's streets and Avenues, Hermann Minkovski defined the distance between A = (x1, y1) and B = (x2, y2) as abs(x1-x2) + abs(y1-y2). In terms of Manhattan, this was equivalent to x1-x2 streets up or down) and (y1-y2) avenues across. Again, the 3-dimensional version is simply an extension of this metric.