User:Brews ohare/Shortest distance

Shortest distance

See also: Euclidean shortest path, Shortest path problem, and Geodesic

Pythagoras' theorem plays a fundamental role in establishing the nature of a space. For example, it is useful to establish the commonly used assertion:

The shortest distance between two points is a straight line

How is this statement related to the fundamental postulates of geometry? That is question requiring some statement of the fundamental postulates. Here is a possible formulation:

Axiom 1: The incidence axiom (straightedge axiom)

(a) ℘ and ʆ are sets; an element of ʆ is a subset of ℘.

(b) If P and Q are distinct elements of ℘, then there is a unique element of ʆ that contains both P and Q.

(c) There exist three elements of ℘ not all in any element of ʆ

Introduction of distance d:

Axiom 2: Ruler postulate

d:℘ × ℘ → ℝ , d: (P, Q) ←→ PQ is a mapping such that for each line l there exists a bijection f:l→ℝ, f:P←→f(P) where

PQ = |f(Q)−f(P)|

for all points P and Q on l.

Mapping d is the distance function and PQ is the distance from point P to point Q. If for line l f:l→ℝ is such that PQ=|f(Q)−f(P)| for all points P and Q in l, then f is a coordinate system for l and f(P) is a coordinate for P with respect to l and f.

Theorem: If P and Q are points, then

$PQ\geqq 0\ .$

$PQ=0\ \mathrm {iff} \ P=Q\ .$

$PQ=QP\ .$

An additional axiom is required to establish the triangle inequality:

$PQ+QR\geqq PR\ .$

Example: Euclidean distance function

$d(P,Q)={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}\ .$

Example: Taxicab geometry:

$t(P,Q)=|x_{2}-x_{1}|+|y_{2}-y_{1}|\ .$

Both satisfy the triangle inequality.