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What is Distance?
A mathematical description of the space between two objects involves some kind of
metric, or a way to functionally describe what it means for objects to be "close
to one another," or "far away."
The general mathematical case for distance is defined in this way. For any set
M, there is a function d: MxM → R, where R stands in for the set of real numbers
that satisfies these three conditions.
1.) The distance d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. In other
words, the distance between two points is always positive, and zero is a point,
not a distance.
2.) The distance d(x,y) = d(y,x). That is to say the distance relationship is
symmetric. The ordering of point we pick should not have an effect on the
3.) The distance satisfies the triangle inequality, or d(x,z) ≤ d(x,y) + d(y,z).
All distances are the shortest path between two points.
In mathematics, this distance function is defined as a metric. Along with the
set M, this metric function defines a metric space for distance.
For Euclidean geometries, the distance between two points in defined in the
following way. Assuming we have a three-dimensional coordinate system with axes
x, y, and z, we can geometrically describe the distance between the two points
(x1, y1, z1) and (x2, y2,
z2) with the function:
d = sqrt((x2-x1)2 + (y2-y1)2 + (z2-z1)2)
The formula above is derived from the
Pythagorean theorem. It can be geometrically
derived by constructing a right triangle with of
its legs on the hypotenuse of another triangle
with its other leg orthogonal to the plane that
contains the first triangle. The length of the
hypotenuse, c, will be equal to the distance
between the two objects via the equation c2 = a2
This mathematical definition of distance should not be confused with other kinds
of distance such as directed distance and displacement (aka distance along a
path). A directed distance is a distance with a direction, or a sense (vector
quantity). Along a straight line, a directed distance is a vector joining any
two points in a Euclidean vector space. Assuming we start at a point A and end
at a point B, the directed distance would be the vector joining point A to point
Displacements are a special kind of directed distance, useful in mechanics. They
are essentially the minimum distance along some kind of path that a particle
travels, not the distance between the point where the particle begins and where
it ends. For example, a particle could travel in a complete circle and have a
very large displacement but have no distance between the point where it began
and the point where it ended.