Definition:Family of Curves

Definition

A family of curves is a set of curves which are described with a common equation, in such a way that all such curves can be generated by varying one or more parameters.

Parameter

The parameters of a family of curves $\FF$ is a set of real numbers which, when varied, generate all the elements of $\FF$.

Classification

One-Parameter Family

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.

Examples

Circles with Centers along $x$-Axis

Consider the equation:

$(1): \quad \paren {x - h}^2 + y^2 = a^2$

where $x, y, a, h \in \R$.

$(1)$ defines a family of circles:

whose radii are determined by the parameter $a$

whose centers are on the $x$-axis of a Cartesian plane at $\tuple {h, 0}$ determined by values of the parameter $h$.

Circles in Plane

Consider the equation:

$(1): \quad \paren {x - h}^2 + \paren {y - k}^2 = a^2$

$(1)$ defines a family of circles:

whose radii are determined by the parameter $a$

whose centers are at $\tuple {h, k}$ on Cartesian plane, determined by values of the parameters $h$ and $k$.

Hence $(1)$ represents the family of all circles in the plane.

Also see

• Definition:Confocal Conics

• Results about families of curves can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): family: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): family: 1.