Parameterization of a curve

Definition

Parameterization of a curve involves expressing the coordinates of the points on the curve as functions of a single variable, known as the parameter. This technique is useful for describing curves that are not easily represented by standard Cartesian equations.

5 Must Know Facts For Your Next Test

1. A curve in the plane can be parameterized by two functions, $x(t)$ and $y(t)$, where $t$ is the parameter.
2. The parameterization is not unique; different choices of the parameter can describe the same curve.
3. To find the length of a parameterized curve from $t=a$ to $t=b$, use the formula: $$L = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$$.
4. The slope of a tangent to a parameterized curve at any point is given by $$\frac{dy/dt}{dx/dt}$$ provided that $dx/dt \neq 0$.
5. Parameterizations are particularly useful for describing motion, where $x(t)$ and $y(t)$ represent positions at time $t$.

Review Questions

Related terms

Parametric Equations:

Equations that express coordinates as functions of one or more independent variables called parameters.

Arc Length:

A measure of the distance along a curved line; for parametric curves, it is calculated using an integral involving derivatives with respect to the parameter.

$\textbf{Polar Coordinates}$: $A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.$