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Tuyetdong Phanyamada

Deposit #145

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More Curves in GeoGebra

The Hypocycloid Family

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Tuyetdong Phanyamada
CSULA 2015       

The following animations offer a comparison of polar and Cartesian graphing.


Graphing calculators and computer software permit construction of complicated graphs within seconds.  However, one must often take a polar equation and write it in parametric form.   If  your grapher does not have a built-in polar graphing command, then take the polar equation and write as follows:


In GeoGebra it is easy to alter all variables and constant terms while watching the dynamically changing curve.  Finally, it is possible to watch an animation using color.

These curves are created with GeoGebra, a free software that allows users to work with Algebra, Geometry, Trigonometry, Statistics and Calculus.  They are graphs of polar functions of the form

where a, b, and c are real numbers and i, k, and m are positive integers.  Some of these graphs are not obtainable on hand-held graphing calculators with lower resolution displays.  Slides 7 and 8 are rotated 90 degrees counterclockwise from the original graphs.

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Tuyetdong Phan-Yamada and Walter M. Yamada III, Exploring Polar Curves with GeoGebra, Mathematics Teacher,  October, 2012, vol. 106 (3), p. 228.
Tuyetdong Phan-Yamada and Elysus Gwin, Hypocycloids and Hypotrochoids, MathAMATYC Educator,  September, 2014, vol. 6 (1), p. 5.

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Hipparchus (ca. 190-120 BC) may have been the first to explore polar geometry.
Hipparchus Stamp
Today we credit him with solving spherical triangles and calculating early trig tables.
He also introduced the division of a circle into 360 degrees into Greece.
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