Cardioids as envelopes (The cardioid in your coffee cup, part 2)
Channel: Kevin's Mathematical Musings
Date: 2021-04-05

Showing how cardioids envelopes of pencils of lines, circles, and other cardioids.

00:00 Introduction
00:52 Review: pencils and envelopes
01:30 Envelope of a pencil of circles
04:44 Implicit Cartesian formula of a cardioid
06:27 Envelope of pencil of circles is a cardioid
06:47 String art in the circle
07:34 String art and coffee cup are the same figure
09:00 String art forms a cardioid
15:36 Envelope of normals to the cardioid (evolute)
16:50 Cardioidal coordinate system
18:03 Summary
18:25 Coming attractions: the Mandelbrot cardioid


IMAGES

“Caïn venant de tuer son frère Abel, by Henri Vidal”, Alex E. Proimos.
https://commons.wikimedia.org/wiki/File:Paris_Tuileries_Garden_Facepalm_statue.jpg
Creative Commons Attribution 2.0 Generic
https://creativecommons.org/licenses/by/2.0/

“Caustique”, Gérard Janot
https://commons.wikimedia.org/wiki/File:Caustique.jpg
GNU Free Documentation License, Version 1.2
https://en.wikipedia.org/wiki/en:GNU_Free_Documentation_License

“Lightbulb with halo”  pnx
https://openclipart.org/detail/202667/lightbulb-with-halo
Creative Commons Universal Public Domain Dedication (CC0 1.0)
https://creativecommons.org/publicdomain/zero/1.0/

“Red Ball” chris_sharkot (altered)
https://openclipart.org/detail/3738/ball
Creative Commons Universal Public Domain Dedication (CC0 1.0)
https://creativecommons.org/publicdomain/zero/1.0/

MUSIC

“Almost New” Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 3.0 License
http://creativecommons.org/licenses/by/3.0/