Polar Coordinates: Example 11: Limaçons + Cochlea + Archimedean Screw

in Threespeak20 days ago

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In this video I go over another example on graphing Polar Coordinates and this time look at the family of curves r = 1 + c sin ϴ which are known as Limaçons, which is a French word which translates to “cochlea” which is derived from the Greek and Latin words for “snail shell” or “screw”. This gets the name Limaçons because the curves do in fact look somewhat like a snail’s shell, for some values of the constant c. I once again use the amazing Desmos calculator to illustrate how the shape changes as we change c. The shapes change from a circular shape with a loop inside it for large values of c, to a cardioid at c = 1, then to a circle at c = 0; and a mirror image of the results for negative values of c. I may go over an “analytic” proof to these properties in a later video so #StayTuned!

Also in this video I take a side quest into learning more about the “cochlea” which is a snail-like structure as part of the human inner ear. It is filled with fluid and thousands of hair cells which transfer the sound vibrations as nerve signals to the human brain. What’s interesting is that prior to the 17th century, the “cochlea” was used in reference to any spiral objects such as a spiral staircase and the famous Archimedean screw. The Archimedean screw (or Archimedes’ screw) is a very interesting piece of engineering ingenuity and comprises of a screw like structure that moves fluid or other objects across the length of the screw by rotating it. I also get side tracked in this video and discuss “dimples” which are indentations on the human body, like dimples on a smile, so hope you enjoy the random topics I cover!

Anyways like always, play around with the Desmos calculator and let me know what other amazing shapes you can discover!

You can play around with the graph that I created in my video here: https://www.desmos.com/calculator/mhxsxtyt5p

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvROaAz3hEliI-MHpQ

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-polar-coordinates-example-11-limacons-cochlea-archimedean-screw

Related Videos:

Polar Coordinates: Example 10: Graphing r = sin(8ϴ/5): https://youtu.be/m-A35NDX7b0
Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk
Polar Coordinates: Example 9: Cardioid: Part 4: https://youtu.be/R6UtQD8tUF0
Polar Coordinates: Example 9: Cardioid: Part 3: Question B: https://youtu.be/-TZHaTjHfb8
Polar Coordinates: Example 9: Cardioid: Part 2: Question A: https://youtu.be/gUYmzZ8Cfhg
Polar Coordinates: Example 9: Cardioid: Part 1: Slope Formula & Trigonometric Algebra: https://youtu.be/MlIhueP36EE
Polar Coordinates: Tangents to Polar Curves: https://youtu.be/DJso0gmxCBY
Polar Coordinates: Symmetry: https://youtu.be/TWxforTOvxM
Polar Coordinates: Example 8: Four-Leaved Rose: https://youtu.be/mTk5j0jD3dE
Polar Coordinates: Example 7: Cardioid: https://youtu.be/rPErcaqNUIY
Polar Coordinates: Example 6: Part 2: Polar Circle to Cartesian: https://youtu.be/biUHN-BphkE
Polar Coordinates: Example 5: Straight Lines: https://youtu.be/AGSz3EVi05A
Polar Coordinates: Example 4: Circle: https://youtu.be/ebsaIlXaZxs
Polar Coordinates: Example 2: Polar to Cartesian: https://youtu.be/mvpb9QshNNU
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: Example 1: https://youtu.be/q_kpqPpoLqE
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw .


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I don’t always go graph Limaçons but when I do I usually get sidetracked and look up the Cochlea, Archimedean screw, and even dimples ;)

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-polar-coordinates-example-11-limacons-cochlea-archimedean-screw