Cissoid of Diocles

Definition

The Cissoid of Diocles is a type of plane curve historically used for solving the problem of doubling the cube. In polar coordinates, its equation can be given as $r = 2a \sin\theta \tan\theta$.

5 Must Know Facts For Your Next Test

1. The Cissoid of Diocles is defined by the equation $r = 2a \sin\theta \tan\theta$ in polar coordinates.
2. It was originally used to solve the problem of duplicating the cube, an ancient Greek mathematical challenge.
3. The curve has a cusp at the origin and extends infinitely in one direction.
4. When parameterized, it can be expressed as $(x,y) = (t^2, t(2a - t^2))$ where $t$ is a parameter.
5. The area under one arch of the cissoid can be calculated using definite integrals.

Review Questions

• What is the polar equation for the Cissoid of Diocles?
• For what historical problem was the Cissoid of Diocles originally used?
• How can you express the Cissoid of Diocles using parametric equations?

Related terms

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

Parametric Equations: Equations that express coordinates as functions of one or more parameters instead of as implicit relationships between variables.

Area Under a Curve: A concept in integral calculus representing the total area between a curve and the x-axis over a given interval.