Cardioid

Definition

A cardioid is a heart-shaped curve generated by tracing a point on the circumference of a circle that is rolling around another circle of the same radius. It can be expressed in polar coordinates as $r = 1 - \cos(\theta)$ or $r = 1 + \cos(\theta)$.

5 Must Know Facts For Your Next Test

1. The standard polar equation for a cardioid is $r = a(1 - \cos(\theta))$ or $r = a(1 + \cos(\theta))$, where 'a' is the radius.
2. A cardioid has one cusp and exhibits symmetry about the x-axis.
3. It can also be represented in parametric form, with equations derived from its polar form.
4. In Cartesian coordinates, converting the polar form involves using $x = r\cos(\theta)$ and $y = r\sin(\theta)$.
5. The area enclosed by a cardioid given by $r = 1 - \cos(\theta)$ is $3\pi/2$.

Review Questions

• What is the general polar equation of a cardioid?
• Describe the symmetry and key features of a cardioid.
• How do you convert the polar form of a cardioid into Cartesian coordinates?

Related terms

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

Limaçon: A limaçon is a type of curve represented by the equation $r = b + a \cos(\theta)$, which can generate various shapes including circles and cardioids.

Parametric Equations: Equations that express the coordinates of points on a curve as functions of a parameter, often used to represent curves like ellipses, circles, and cardioids.