# Definition:Parametric Equation

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## 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**.

The set:

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

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

### Parameter

$t$ is referred to as the **(independent) parameter** of $\set {\phi_k: 1 \le k \le n}$.

### $2$ Dimensions

Definition:Parametric Equation/2 Dimensions

## 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.

## Also see

- Results about
**parametric equations**can be found**here**.

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(1)$ Gradient forms