Definition:One-Parameter Family

Definition

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One-Parameter Family of Curves

Consider the implicit function $\map f {x, y, c} = 0$ in the cartesian $\tuple {x, y}$-plane where $c$ is a constant.

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

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

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

One-Parameter Family of Surfaces

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.

Also see

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