I found a page in one of my notebooks describing a particular class of 'surfaces' which are given by the implicit equations:
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}$$
Or, more generally
$$\sum_i \left(a_ix_i+b_i\right)^{-1}=\left(\sum_ia_ix_i+b_i\right)^{-1}\quad:\quad\mathbf{x}\in\mathbb{R}^n$$
I only have notes on $\mathbf{x}\in\mathbb{R}^2,\mathbb{R}^3$ and I'm not sure what the context was at the time I wrote it, but it looks interesting enough and I would like to know more.
Do these objects have a name? Have they been studied before?
Edit:
After working on it for a bit, it occurred to me that in three dimensions this is the same as
$$(u_1+u_2)(u_1+u_3)(u_2+u_3)=0\quad:\quad u_i=a_ix_i+b_i$$
Which is a particular case of
$$({u_1}^n+{u_2}^n)({u_1}^n+{u_3}^n)({u_2}^n+{u_3}^n)=0$$
For whatever reason, this last bit seems incredibly familiar. I'm not sure why, but it brings to mind something about quaternions, vector norms, and general relativity. Where have I seen this before?