Definition:Parametric Equation
(Redirected from Definition:Freedom-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