# Difference between revisions of "Line"

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
This page is not of significant importance, but thinking about it I have no formal definition of a line, how it pertains to linear maps ect, even though it's "obvious" in my head, Remember:
• We must be able to account for lines in "weird" spaces, like hyperbolic lines, (arguably) lines in polar coordinates, like [ilmath]r(\theta)\eq 1[/ilmath] giving a circle.
Alec (talk) 13:00, 14 December 2018 (UTC)
[ilmath]\newcommand{\v}{\mathbf{#1} } [/ilmath]
See also: line (terminology) for an overview of line vs (line) segment vs ray

## Definition

### Parametrised line

Let [ilmath]\v a\in\mathbb{R}^n[/ilmath] be given (for some [ilmath]n\neq 0[/ilmath]) and let [ilmath]\v b\in\big(\mathbb{R}^n-\{\v 0\}\big) [/ilmath] be given also[Note 1]

• Define [ilmath]\ell:\mathbb{R}\rightarrow\mathbb{R}^n[/ilmath]Warning:We may be able to relax these[Note 2], we will use [ilmath]t\in\mathbb{R} [/ilmath] for the parameter to the function [ilmath]\ell[/ilmath]
• Where [ilmath]\ell[/ilmath] is given by: [ilmath]\ell:t\mapsto \v a +t\cdot\v b[/ilmath]

If we take a point [ilmath]\v c\in\mathbb{R}^n[/ilmath] in addition to [ilmath]\v a[/ilmath] (taken above) if we define:

• [ilmath]\v b:\eq \v c-\v a[/ilmath] then

[ilmath]\ell(t)[/ilmath] is at [ilmath]\v a[/ilmath] for [ilmath]t\eq 0[/ilmath] and moves at a uniform speed towards [ilmath]\ell(t)\eq \v c[/ilmath] when [ilmath]t\eq 1[/ilmath], that is to say:

• [ilmath]\ell\big\vert_{[0,1]\subseteq\mathbb{R} }:[0,1]\rightarrow\mathbb{R}^n[/ilmath] (Notation: restriction) yields the line segment from [ilmath]\v a[/ilmath] to [ilmath]\v b[/ilmath]
• Furthermore, this degenerates correctly in the case that [ilmath]\v a\eq \v c[/ilmath] (which would give [ilmath]\v b\eq \v 0[/ilmath]

Restricting [ilmath]t[/ilmath] to [ilmath]\mathbb{R}_{\ge 0} [/ilmath] yields the ray based at [ilmath]\v a[/ilmath] through [ilmath]\v c[/ilmath] (provided we may call the case of [ilmath]\ell[/ilmath] given by [ilmath]\v c\eq \v a[/ilmath] a "ray" at all, as it's a point, same as the segment and line in this case)

Not restricting [ilmath]t[/ilmath] at all, so [ilmath]t\in\mathbb{R} [/ilmath], yields the line through [ilmath]\v a[/ilmath] and [ilmath]\v c[/ilmath], provided again we count the degenerate case [ilmath]\v a\eq \v c[/ilmath] a line.