User:Brews ohare/Shortest distance

Shortest distance
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.