Definition:Family of Surfaces/One-Parameter

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.

Parameter

The value $c$ is the parameter of $F$.

Examples

Concentric Spheres

The equation:

$x^2 + y^2 + z^2 = r^2$

is a one-parameter family of concentric spheres whose centers are at the origin of a Cartesian $3$-space and whose radii are the values of the parameter $r$.

Also see

• Results about one-parameter families of surfaces can be found here.