# Definition:Parametric Equation

## Definition

Let $\mathcal R \left({x_1, x_2, \ldots, x_n}\right)$ be a relation on the variables $x_1, x_2, \ldots, x_n$.

Let the truth set of $\mathrel R$ be definable as:

$\forall k \in \N: 1 \le k \le n: x_k = \phi_k \left({t}\right)$

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:

$x_k = \phi_k \left({t}\right)$

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

The set:

$\left\{ {\phi_k: 1 \le k \le n}\right\}$

is a set of parametric equations specifying $\mathcal R$.