# 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 $A$. 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.

# Teaching Materials

  [[Description::The Cardioid an epicycloid with one cusp.|]]
`
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 12, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8766779899597168, "perplexity": 883.8291092644237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711111.35/warc/CC-MAIN-20221206161009-20221206191009-00060.warc.gz"}