Is it possible to index the solutions to Pythagoras equation if we let $ x,y,z$ be linear forms? In other worse I’m asking for solutions to:

$ $ \big(a_1 s + a_2 t\big)^2 + \big(b_1 s + b_2t \big)^2 = \big(c_1 s + c_1 t \big)^2 $ $

Is it just these same as the formula over integers or rationals?

\begin{eqnarray*} x &=& m^2 – n^2 \ y &=& 2mn \ z &=& m^2 + n^2 \end{eqnarray*}

We could intersect lines of rational slope through $ (0,1)$ the circle $ S^1 = \{ x^2 + y^2 = z^2\}$ . If group like terms in the original equation I get something very different:

\begin{eqnarray*} a_1^2 + b_1^2 &=& c_1^2 \ a_1 a_2 + b_1 b_2 &=& c_1 c_2 \ a_2^2 + b_2^2 &=& c_2^2 \end{eqnarray*}

Hopefully this is a variety and not just a scheme, but either way suggests we can solve these equations over $ \mathbb{Z}$ . Is this a homogenous space?