Definition:Family of Curves/One-Parameter

Definition

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.

Parameter

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

Examples

Concentric Circles

The equation:

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

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

Circles of Equal Radius with Centers along $x$-Axis

Consider the equation:

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

where $a$ is constant.

$(1)$ defines a one-parameter family of circles of constant radius $a$ whose centers are on the $x$-axis of a Cartesian plane at $\tuple {h, 0}$ determined by values of the parameter $h$.

Sources

• 1956: E.L. Ince: Integration of Ordinary Differential Equations (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions: $(1.1)$
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions
• 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.