Cardioid

Definition

A cardioid is a heart-shaped curve described by the polar equation $r = a(1 + \cos\theta)$ or $r = a(1 + \sin\theta)$. It is a special type of limaçon and is symmetric about the x-axis or y-axis depending on its form.

5 Must Know Facts For Your Next Test

1. A cardioid can be generated by tracing a point on the circumference of a circle as it rolls around another circle of the same radius.
2. The Cartesian coordinates for points on the cardioid given by $r = a(1 + \cos\theta)$ are $(a(2\cos^2\frac{\theta}{2}), a(2\cos^2\frac{\theta}{2}) - 2a)$.
3. The area enclosed by the cardioid $r = a(1 + \cos\theta)$ is $3\pi a^2/2$.
4. The length of the arc of one loop of the cardioid $r = a(1 + \cos\theta)$ is $8a$.
5. Cardioids exhibit cusp singularities at their vertices, where the derivative with respect to $\theta$ becomes undefined.

Review Questions

• What is the polar equation for a cardioid centered at the origin?
• How do you calculate the area enclosed by the cardioid $r = 3(1 + \sin \theta)$?
• What characteristic point does every cardioid have that differentiates it from other limaçons?

Related terms

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

Limaçon: A family of curves defined by polar equations of the form $r = b + a \cos (k \theta)$, which includes circles, ellipses, and cardioids.

Parametric Equations: Equations that express coordinates of points as functions of one or more parameters.