# Definition:Parametric Equation

## Definition

Let $\map \RR {x_1, x_2, \ldots, x_n}$ be a relation on the variables $x_1, x_2, \ldots, x_n$.

Let the truth set of $\RR$ be definable as:

$\forall k \in \N: 1 \le k \le n: x_k = \map {\phi_k} t$

where:

$t$ is a variable whose domain is to be defined
each of $\phi_k$ is a mapping whose domain is the domain of $t$ and whose codomain is the domain of $x_k$.

Then each of:

$x_k = \map {\phi_k} t$

is a parametric equation where $t$ is the parameter.

The set:

$\set {\phi_k: 1 \le k \le n}$

is a set of parametric equations specifying $\RR$.

## Also known as

Some older texts, particularly in the context of analytic geometry, refer to such equations as freedom-equations, as they express the freedom of the movement of the tuple $\tuple {x_1, x_2, \ldots, x_n}$ as $t$ changes.