# Difference between revisions of "Cardioid"

Cardioid
Field: Geometry
Image Created By: Wolfram Math World

Cardioid

The Cardioid an epicycloid with one cusp.

# Basic Description

The image is formed by tracing a point on the circumference of a circle of radius $a$, without slipping, on another stationery circle, as in the image below

.

# A More Mathematical Explanation

The curve is given by:

• Cartesian equation $({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})$, where '"UNIQ--math-00 [...]

The curve is given by:

• Cartesian equation $({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})$, where $a$ is the radius of the moving circle.
• Polar equation $r = a (1 - {\cos} {\theta})$
• Parametric equation

$x = a {\cos} t (1 - {\cos t})$

$y = a {\sin} t (1 - {\cos t})$

## Properties

• It has a cusp at the origin.
• There are exactly three tangents to the cardioid with any given gradient
• The tangents at the ends of any chord through the cusp point are at right angles
• The length of any chord through the cusp point is 2$a$

## Generating a cardioid

Draw a circle $C$, and pick a fixed point $A$ on it. Then, draw a set of circles centered on the circumference of $C$ and passing through $A$lo

# Teaching Materials

  [[Description::The Cardioid an epicycloid with one cusp.|]]
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