Knowlpedia

Line Connecting Two Points

The affine line through a and b: {λa+(1−λ)b : λ∈R}.

Let XX be a and let a,b∈Xa,b\in X. The line connecting aa and bb is

L[a,b]:={λa+(1−λ)b∣λ∈R}.L[a,b]:=\{\lambda a+(1-\lambda)b\mid \lambda\in\mathbb{R}\}.

This is the smallest containing {a,b}\{a,b\} and is the affine analogue of the [a,b][a,b] (where λ∈[0,1]\lambda\in[0,1]).

Examples
  • In R2\mathbb{R}^2, L[a,b]L[a,b] is the usual straight line through aa and bb.
  • If a=ba=b, then L[a,a]={a}L[a,a]=\{a\}.