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

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Published: 23 May 2017 › Updated: 23 May 2017Polar Coordinates: Example 11: Limaçons + Cochlea + Archimedean Screw

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

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 PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIhvROaAz3hEliI-MHpQ

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

Polar Coordinates Example 11.jpeg

Example 11

Investigate the family of polar curves given by r = 1 + c sin θ.

How does the shape change as c changes?

These curves are called Limaçons, after a French word for snail, because of the shape of the curves for certain values of c

https://translate.google.ca/?rlz=1C1CHBF_enCA730CA730&um=1&ie=UTF-8&hl=en&client=tw-ob#fr/en/lima%C3%A7on
Retrieved: 22 May 2017
Archive: https://archive.is/chDxJ

https://mechanicalcochlea.wordpress.com/home/the-cochlea/
Retrieved: 22 May 2017
Archive: https://archive.is/EzYD5

The Conchlea

What is the Cochlea?

The cochlea is an integral part of hearing. The human ear is separated into three main parts: The outer ear, the middle ear, and the inner ear. The outer ear contains the visible part of the ear and moves along the channel collecting sound waves and passing them to the middle ear. At the middle ear the sound waves hit the tympanic membrane (ear drum) which vibrates the three tiny ear bones. These bones react and transfer the sound into a wave vibration that is transferred to the cochlea. The cochlea is part of the inner ear, and is described as a bony structure filled with fluid and thousands of hair cells that react to the vibrations from the middle ear. The hair cells are each linked to a certain frequency and when that frequency is made, the hair cell sends a nerve signal to the brain. Figure 1 shows a representation of the parts of the ear. The cochlea is located on the right past the three ear bones.

image.png

Figure 1:

As seen in figure 1, the cochea is a snail like structure, with an unrolled length of 3cm.

https://www.google.ca/search?q=define%3Acochlea
Retrieved: 22 May 2017
Archive: https://archive.is/OYPhV

coch·le·a
ˈkōklēə,ˈkäklēə/

noun
noun: cochlea; plural noun: cochleae

  1. The spiral cavity of the inner ear containing the organ of Corti, which produces nerve impulses in response to sound vibrations.

mid 16th century (used to denote spiral objects such as a spiral staircase and an Archimedean screw): from Latin, ‘snail shell or screw,’ from Greek kokhlias . The current sense dates from the late 17th century.

https://en.wikipedia.org/wiki/Archimedes%27_screw
Retrieved: 22 May 2017
Archive: https://archive.is/JJKLm

Archimedes' screw

The Archimedes screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches. Water is pumped by turning a screw-shaped surface inside a pipe.

The screw pump is commonly attributed to Archimedes on the occasion of his visit to Egypt. This tradition may reflect only that the apparatus was unknown to the Greeks before Hellenistic times and was introduced in Archimedes's lifetime by unknown Greek engineers.[1] Some writers have suggested the device may have been in use in Assyria some 350 years earlier[citation needed].

The Archimedes screw was operated by hand and could raise water efficiently

An Archimedes screw in Huseby south of Växjö Sweden

Archimedes screw

Modern Archimedes screws which have replaced some of the windmills used to drain the polders at Kinderdijk in the Netherlands

Archimedes screw as a form of art by Tony Cragg at 's-Hertogenbosch in the Netherlands

An Archimedes screw seen on a combine harvester

Solution:

The figures below show the graphs for various values of c.

https://www.desmos.com/calculator/mhxsxtyt5p

image.png


Retrieved: 23 May 2017
Archive: https://archive.is/wafFT

image.png

For c > 1 there is a loop that decreases in size as c decreases.

When c = 1, the loop disappears and the curve becomes the cardioid that we sketched in Example 7.

For c between 1 and 1/2 the cardioid's cusp is smoothed out and becomes a "dimple".

https://en.wikipedia.org/wiki/Dimple
Retrieved: 22 May 2017
Archive: https://archive.is/H9TnG

Dimple

A dimple (also known as a gelasin[1]) is a small natural indentation in the flesh on a part of the human body, most notably in the cheek or on the chin.

…

Cheek dimples when present, show up when a person makes a facial expression. A chin dimple is a small line on the chin that stays on the chin without making any facial expressions. Dimples may appear and disappear over an extended period.[2]


Retrieved: 22 May 2017
Archive: https://archive.is/uJL2i

image.png

When c decreases from 1/2 to 0, the Limaçon is shaped like an oval.

This oval becomes more circular as c → 0, and when c = 0 the curve is just the circle r = 1.

The remaining parts of the above figures show that as c becomes negative, the shapes change in reverse order.

In fact these curves are reflections about the horizontal axis of the corresponding curves with positive c.

Note: I may go over an analytic proof of the different types of shapes formed by changing the value of c, in a later video, so #StayTuned!

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