Cardioid
Cardioid |
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Cardioid
- The Cardioid an epicycloid with one cusp.
Contents
Basic Description
The image is formed by tracing a point on the circumference of a circle of radius , without slipping, on another stationery circle, as in the image below
.
A More Mathematical Explanation
The curve is given by:
- Cartesian equation , where '"`UNIQ--math-00 [...]
The curve is given by:
- Cartesian equation , where is the radius of the moving circle.
- Polar equation
- Parametric equation
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
Generating a cardioid
Draw a circle , and pick a fixed point on it. Then, draw a set of circles centered on the circumference of and passing through . The envelop of the chords of these circles is a cardioid, as in the main image.
An instance where one could see a cardioid is when looking into a cup of coffee. The caustic seen at the bottom of a cup of coffee could be a cardioid, depending on the angle of light relative to the bottom of the cup.
Also, all unidirectional microphones are cardioid-shaped.
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[[Description::The Cardioid an epicycloid with one cusp.|]]