Name of a set of points given by an equation

Question

If $f:\mathbb R^2\to\mathbb R$ is a function, then we call the set of points $(x,y,f(x,y))$ the graph of the function $f$. When we can't explicitly define a function in a given equation, is there a name for the set of points given by a relation? In another words, is there a name for the set of the points given implicitly by an equation? for example, the points $(x,y)\in \mathbb R^2$ such that $x^2+y^2=1$.

Answer 1

Given an equation $f(x_1, \ldots, x_n) = 0$, in a geometric context we call the set $$\{(x_1, \ldots, x_n) : f(x_1, \ldots, x_n) = 0\}$$ the locus of solutions to the equation. (The plural is irregular, loci.) This set is also sometimes called the zero locus of the function $f$.

Note that this is qualitatively different from the notion of graph of a function: For a function $f$, the locus of solution of $f(x_1, \ldots, x_n) = 0$ is a subset of $\mathrm{dom}(f)$, whereas the graph of $f$ is a subset of $\mathrm{dom}(f) \times \mathrm{im}(f)$.