## What is there in the word “distance”?

In Education, Mathematics on February 6, 2013 at 9:32 PM

I fist learned about distances in school- class of classical geometry. Of course it was not called classical geometry in school but only geometry. The initial concepts were related to distances between points in a plane, between lines in a plane and between a line and a point in a plane. A plane, as you know, is a  2-dimensional (2D) space. In high school, this concept was extended to 3-dimensional space (3D). The concept of distance basically gives an idea of how far (or how close) are two things (lines, points) from (to) each other. What I learned was “2-norm distance” (the typical Euclidean distance):

$2-norm distance = \sqrt{(\displaystyle\sum_{i=1}^{n} |{x_{i}}^2 - {y_{i}}^2|)}$

I learned about Hamming Distance during my undergraduate courses on electronics communication. However, it is only during research that I learned a lot more about distances. My first surprise came when I heard about distances in a class on image processing. You can use distances to measure similarity between images! Of course the definitions and methods to calculate those distances were also different. Since then I have learned about distances being one major way of identifying similarities between objects or classes of objects. The central idea behind all these different kinds of distances (not just in image processing) remains the same: to measure  how far the objects are from each other in some respect. For instance, in psychoanalysis “Emotional Distance” is the degree of emotional detachment from some person or events; Czekanovski-Dice distance is used to compare two audio waveforms x and y in time domain etc. If your distance from the world of distances is not big ;), you might want to try reading the Dictionary of Distances.