Parametric curve

Definition

A parametric curve is a set of points defined by parametric equations where each coordinate is expressed as a function of one or more parameters. Typically, in two dimensions, these equations take the form $x(t)$ and $y(t)$, where $t$ is the parameter.

5 Must Know Facts For Your Next Test

1. Parametric curves are often used to describe motion and trajectories where position depends on time.
2. The derivative of the parametric equations gives information about the slope and direction of the curve at any point.
3. To find the length of a parametric curve from $t = a$ to $t = b$, use the integral formula: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt.$$
4. Eliminating the parameter can sometimes convert parametric equations into a single equation in $x$ and $y$.
5. Parametric curves can intersect themselves, depending on how the parameterization is defined.

Review Questions

• What are parametric equations and how do they define a curve?
• How can you find the length of a parametric curve?
• Explain how derivatives are used to determine properties like slope for parametric curves.

Related terms

Parameter: An independent variable that defines a family of objects or functions. In parametric equations, it typically represents time or another continuous change.

$\text{Derivative}$: $\text{The rate at which one quantity changes with respect to another. For parametric curves, derivatives help compute slopes and tangents.}$

$\text{Arc Length}$: $\text{The distance along a curved line or surface. For parametric curves, it is computed using an integral involving derivatives of the parameterized functions.}$