Metrics in 2D Analytic Geometry

Metrics in 2D Analytic Geometry
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Definition

A metric is a function, \(d\) that assigns a real number to each pair of points, and that satisfies:
- \(d(\bs{p}, \bs{q}) \geq 0\)
- \(d(\bs{p}, \bs{q}) = 0 \mbox{ iff } \bs{p} = \bs{q}\)
- \(d(\bs{p}, \bs{q}) = d(\bs{q}, \bs{p})\)
- \(d(\bs{p}, \bs{q}) \leq d(\bs{p}, \bs{r}) + d(\bs{r}, \bs{q})\) (the triangle inequality)
A Metric Generated by the Euclidean Norm:
- \(d(\bs{p}, \bs{q}) = ||\bs{p} - \bs{q}|| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}\)

The Triangle Inequality

Two Cases for the Euclidean Distance

Other Norms

The Rectilinear/Manhattan Norm on \(\mathbb{R}^{2}\):
- \(||\bs{a}|| _{1} = |a_{1}| + |a_{2}|\)
- So: \(d_1(\bs{p} - \bs{q}) = |p_{1} - q_{1}| + |p_{2} - q_{2}|\)
The Supremum Norm on \(\mathbb{R}^{2}\):
- \(||\bs{a}||_{\infty} = \max \{|a_{1}| , |a_{2}| \}\)
- So: \(d_{\infty}(\bs{p} - \bs{q}) = \max \{|p_{1} - q_{1}| , |p_{2} - q_{2}| \}\)

Metrics for Different Norms

Distance Between Two Points

The Unit Ball

For Different Norms

Some Unusual Metrics

The Post Office Metric:
- \(d_{P}(\bs{p}, \bs{q}) = ||\bs{p}|| + ||\bs{q}|| \mbox{ if } \bs{p} \neq \bs{q}\) (and 0 otherwise)
The Radar Screen Metric:
- \(d_{R}(\bs{p}, \bs{q}) = \min \{1, ||\bs{p}-\bs{q}|| \}\)

Some Unusual Metrics