N-plane

Stub grade: A

This page is a stub

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 requires some pictures or at least discussion, the constraint that $\sum^N_{i=0}t_i=1$ is interesting. It doesn't require that they're positive so things like $t_0=100$ and $t_1=-99$ are possible.

Note: Capital-$N$ is used on this page because the $N$-plane is constructed from a set of $N+1$ points - of $\mathbb{R}^n$ - be careful with this distinction.

Definition

Given a geometrically independent set of points of $\mathbb{R}^n$, $\{a_0,\ldots,a_N\}$ we define the $N$-plane, $P$ spanned by these points to be the set:

• $$\left\{x\in\mathbb{R}^n\ \left\vert\ x=\sum^N_{i=0}t_ia_i\wedge \sum_{i=0}^Nt_i=1\text{ for }t_i\in\mathbb{R}\right\}\right.$$

It can also be described as the set of points:

• $$\left\{x\in\mathbb{R}^n\ \left\vert\ x=a_0+\sum^N_{i=1}t_i(a_i-a_0)\text{ for }t_i\in\mathbb{R}\right\}\right.$$

Notes for development of the page

Clearly from the second definition the plane is infinite in size, so the sum of t being 1 constraint must play some part in keeping it going through $a_0$. I also need to prove the claim that these are indeed equivalent.