{"id":145,"date":"2013-10-01T08:03:42","date_gmt":"2013-10-01T08:03:42","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=145"},"modified":"2015-07-29T00:54:40","modified_gmt":"2015-07-29T00:54:40","slug":"catacaustic-of-a-cardioid","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/10\/01\/catacaustic-of-a-cardioid\/","title":{"rendered":"Catacaustic of a Cardioid"},"content":{"rendered":"
\n
\"Catacaustic<\/a>

Catacaustic of a Cardioid – Greg Egan<\/p><\/div>\n<\/div>\n

This image, drawn by Greg Egan<\/a>, shows a cardioid and its catacaustic.<\/p>\n

The cardioid<\/a> is a heart-shaped curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The catacaustic<\/a> of a curve in the plane is the envelope of rays emitted from some source and reflected off that curve.<\/p>\n

If we shine rays from the cusp of the cardioid, the resulting catacaustic is a curve called the nephroid<\/a>. This is the curve traced by a point on the perimeter of a circle that is rolling around a fixed circle whose radius is twice as big!<\/i><\/p>\n

Does this pattern continue? What is the catacaustic of a nephroid? And in general, given an algebraic curve of degree $d$, what can we say about the degree of its catacaustic? If you get stuck on these puzzles, see:<\/p>\n

\u2022 Greg Egan, Catacaustics, resultants and kissing conics<\/a>.<\/p>\n

For more on cardioids, nephroids and other curves formed by rolling one circle on another, see:<\/p>\n

\u2022 John Baez, Rolling circles and balls<\/a>.<\/p>\n

It’s worth mentioning that we can get the nephroid as a catacaustic in another way<\/a>, too: by shining light on a circle from a point at infinity, and letting the rays bounce off the inside of the circle.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image, drawn by Greg Egan, shows a cardioid and its catacaustic. The cardioid is a heart-shaped curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The catacaustic of a curve in the plane is the envelope of rays emitted from some…<\/p>\n

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