MATHEMATICA^{®}Code

For the
student . . . .
In addition to MATHEMATICA@, the Nephroid family of curves
is easily entered and modified on a graphing calculator.


Historical Sketch
The name nephroid means "kidneyshaped"
from the Greek for nephróros meaning "kidney" and éidos
meaning form.
The nephroid was investigated extensively
in the late 17th century. Earlier, an
exchange of ideas among scholars centered on circles and cycloids.
Mersenne, Galileo, Roberval, Descartes, Pascal, and Wallis among others,
contributed to the discussion.
Their successors naturally went on to identify more complicated
plane curves with unique properties. For example,
in 1692 Jakob Bernoulli (1654
 1705) showed that when a light source is placed at the cusp of
a cardioid, the catacaustic, or envelope of the family of reflected rays,
is a nephroid.
"Catacaustics of a circle can be seen as the bright curves
on the surface of coffee in a cup or upon the table inside a curcular
napkin ring." (You can) "observe some catacaustics of a circle using
a cup of liquid and a movable light source."
Howard Eves

Some of the more important names:
Nephroid
Evolute
Nephroid Involute
Catacaustic of a Circle
Catacaustic of a Cardioid
Spherical Nephroid

Some of the more important investigators:
Huygens (1678)
Tschirnhausen (ca. 1679)
Jakob Bernoulli (1692)
Daniel Bernoulli (1725)
Proctor (1878)
Freeth (1879)


Some of the more important historical publications:

Huygens' Traité
de la luminère published in 1690.

T. J. Freeth's paper published
by the London Mathematical Society in 1879.

R. A. Proctor, The Geometry
of Cycloids (London, 1878).

A useful catalog of
curves can be found in older editions of the Encyclopaedia Britannica.
Search under Curves, and/or Special
Curves written by R. C. Archibald.

Useful Links
and Books

http://wwwhistory.mcs.stand.ac.uk/history/Curves/Nephroid.html

http://mathworld.wolfram.com/Nephroid.html

Eves, Howard,
An Introduction to the History of
Mathematics, 6th ed,. The Saunders College Publishing,
1990.

Gray, Alfred,
Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA^{®}, 2nd
ed., CRC Press, 1998, p. 898. 
Lockwood,
E. H., A Book of Curves, Cambridge University
Press, 1961.

Shikin, Eugene V., Handbook
and Atlas of Curves, CRC Press, 1995.

Yates, Robert,
CURVES AND THEIR PROPERTIES, The
National Council of Teachers of Mathematics,
1952.



MATHEMATICA^{®}
Code and animations contributed
by
Gus Gordillo, 2004.


