Definition:One-Parameter Family of Surfaces

Definition
Consider the implicit function $\map f {x, y, z, c} = 0$ in the Cartesian $3$-space where $c$ is a constant.

For each value of $c$, we have that $\map f {x, y, z, c} = 0$ defines a relation between $x$, $y$ and $z$ which can be graphed in cartesian $3$-space.

Thus, each value of $c$ defines a particular surface.

The complete set of all these surfaces for each value of $c$ is called a one-parameter family of surfaces.