# Finding a basis for a set of points in plane and a lattice.

Suppose my domain is $\mathbb{R}^3$ where I have:

• a set of points $A = \left\lbrace x_i \right\rbrace_{i=1}^{n_1}$ and $n_1 > 3$.
• $A$ is contained in a plane (2-dimensional) $P_1$. i.e. $A \subset P_1 \subset \mathbb{R}^3$.
• There is no basis $\left( v_1, v_2, v_3 \right)$ for $A$ $\textit{and}$ a lattice $\Lambda \subset P_1$

I want to construct a mapping, $f: \mathbb{R}^3 \mapsto B$, such that:

• There exists a basis $\left( w_1, w_2, w_3 \right)$ for $f(A)$ $\textit{and}$ $f(A*) \subset f(P_1)$ where $f(A) \cup f(A*)$ is a lattice in $B$. That is to say $$\left(f(A) \cup f(A*) \right) \subset f(P_1) \subset B$$
• I can derive/define $(w_1, w_2, w_3)$ algebraically. i.e. $$\begin{array}{rcl} w_1&=& \text{equation}_1 \in B\\ w_2&=& \text{equation}_2 \in B\\ w_3&=& \text{equation}_3 \in B\\ \end{array}$$
• $f$ is bijective.

How do I do this? I figure $f$ would need to be a piece-wise shearing function where it takes $A$ as input and then $f$ finds some other set in $\mathbb{R}^3$ and the the mapped union of those sets forms a lattice in $B$. Although I am not sure.

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What's $A*{}{}$? – joriki Sep 12 '12 at 17:28
Are there any conditions on $f$ and $B$? At first I assumed $B$ was a vector space and $f$ linear but then $f(A)$ would be at most 2-dimensional. – Simon Markett Sep 13 '12 at 10:19
@joriki $A*$ is an arbitrary collection of points in $P_1$. i.e. $A* \subset P_1$. I believe this does imply the necessity of $f$ being injective when restricted to to the domain $P_1$. – Maelstrom Yamato Sep 14 '12 at 17:03
@SimonMarkett the only condition is $f$ is bijective. $B$ is a codomain. Ideally I would like to say that $B = \mathbb{R}^3$. $f(A)$ need not be 2-dimensional. $f$ may or may not be linear. I would like to derive a function that that constructs a lattice given a collections of points in a plane. The best tool I've seen that may be of use to me is Schinzel's Theorem and mathworld.wolfram.com/KulikowskisTheorem.html – Maelstrom Yamato Sep 14 '12 at 17:16