Cycloid

Definition

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line. It can be described using parametric equations involving trigonometric functions.

5 Must Know Facts For Your Next Test

1. The parametric equations for a cycloid are $x = r(t - \sin(t))$ and $y = r(1 - \cos(t))$, where $r$ is the radius of the rolling circle.
2. Cycloids have cusps, which are points where the curve touches the baseline and has infinite curvature.
3. The arc length of one arch of a cycloid can be computed as $8r$, where $r$ is the radius of the generating circle.
4. The area under one arch of a cycloid is exactly three times the area of the generating circle, calculated as $3\pi r^2$.
5. Cycloids exhibit periodic behavior with each period corresponding to one complete revolution of the generating circle.

Review Questions

• What are the parametric equations that define a cycloid?
• How do you calculate the arc length of one arch of a cycloid?
• What is unique about the cusps in a cycloid?

Related terms

Parametric Equations:

Equations that express coordinates of points on a curve as functions of a parameter, often used to describe complex curves like cycloids.

Arc Length:

The distance measured along the path of a curve. For parametric curves, it involves integrating over the parameter.

Radius: \(r\). A constant distance from any point on a circle to its center, playing a crucial role in defining curves like cycloids.