# Using a single scalar equation to describe a line in space

A line in three-dimensional space may be described as an intesection of two planes, for example: \begin{align}x+y+z=0\tag{1}\\3x+7y=1\tag{2}\end{align} This can be understood as two separate scalar equations or as a single matrix equation. (One may also describe a line parametrically.)

This poses a question: is it possible to express the same line using a single scalar equation? It turns out that it is. The equation $$x^2+y^2=0\tag{3}$$ can be understood as describing the set of all points $(x,y,z)\in\Bbb R^3$ for which $x=0$ and $y=0$. In other words, it describes the $z$-axis.

So, we have described a line in $\Bbb R^3$ using a single scalar equation. But this means any line in $\Bbb R^3$ (or $\Bbb R^n$ for that matter) can be described by a single scalar equation, simply by using an appropriate affine transformation on the equation $(3)$.

As pointed out by Pantelis Damianou in the comment below, this gives us the equation $$(x+y+z)^2+(3x+7y-1)^2=0$$ in the case described above. Note that this tells us exactly the same thing as the equations $(1)$ and $(2)$, since $z^2+w^2=0$ is just another way to say that $z=0$ and $w=0$.

My question is:

Is this point of view ever useful? Does it have any striking applications? Is there an area of mathematics that uses such equations in a fruitful way?

Thanks.

-
So, the first example is $(x+y+z)^2+(3x+7y-1)^2=0$. –  PAD Jul 11 '12 at 21:34
@PantelisDamianou: indeed! I'll add that. Thanks. –  Dejan Govc Jul 11 '12 at 21:36

More generally, we can "pack up" any finite system of equations (over the reals) as a single equation involving a sum of squares. If you want an approximate solution of the system of equations, you can try to minimize that sum of squares: this is the starting point for Least Squares Approximation, which is very useful in data analysis.

-
A set of $D$ of natural numbers is Diophantine if there is a polynomial $P(y,x_1,\dots,x_n)$ such that for all all natural numbers $y$, $$y\in D \quad \text{iff} \quad \exists x_1 \cdots\exists x_n(P(y,x_1,\dots,x_n)=0).$$ Here by tradition the $x_i$ range over the natural numbers, which are defined to include $0$.