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The Spiral Family of Plane Curves:
The Spirals of Archimedes, of Fermat, of
Euler, of Cornu,
 Hyperbolic, Logarithmic,
Spherical, Parabolic, Nielsen's, Seiffert . . . .

The Spiral of
Archimedes 
Polar Equations

Replay the animation 
Def: The spiral is the locus of a point P moving uniformly along
a ray that, in turn, is uniformly rotating in a plane about its origin.
Segment OP is proportional to angle AOP. 
MATHEMATICA^{®}Code
for a Hyperbolic Spiral
The spiral curves are easily entered
and modified on a graphing calculator.

The spiral on the tomb
of Jakob (James) Bernoulli.
Eadem mutata resurgo.
I shall arise the same though
changed. 
Applications
The Spiral of Cornu

An equation for a simple harmonic oscillator
may be dampened. The spiral point at the origin represents
the equilbrium position. The eigenvalues are complex conjugates.

The trace of an underdamped harmonic
oscillator.

The Spiral of Cornu
is named for the French scientist Marie Alfred Cornu (1841  1902).
He studied this curve, also known as a clothoid or Euler's
Spiral, in connection with diffraction. Euler applied a similar
figure while measuring the elasticity of a spring.
The parametric equations for a generalized Cornu spiral
are on the right.
Similar integrals are named for Augustin Jean Fresnel
(17881827), one of the founders of the wave theory of light.


The Spiral of Cornu,
a.k.a. Clothoids "are important curves used in freeway and
railroad construction. For example, a clothoid is needed to make
the gradual transition from a highway; which has zero curvature, to the
midpoint of a freeway exit, which has nonzero curvature. A clothoid
is clearly preferable to a path consisting of straight lines and circles,
for which the curvature is discontinuous." (!!)
Alfred Gray

Other Animations with MATHEMATICA®Code
Historical Sketch
From the legendary Delian problem in antiquity to modern freeway
construction, spirals have attracted great mathematical talent. Among
the more famous are Archimedes, Descartes, Bernoulli, Euler, and Fermat,
but there are many more whose work has enormously influenced pure mathematics,
science and engineering.
The name spiral, where a curve winds outward from a
fixed point, has been extended to curves where the tracing point
moves alternately toward and away from the pole, the socalled sinusoidal
type. We find Cayley's Sextic, Tschirnhausen's Cubic, and Lituus'
shepherd's (or a bishop's) crook. Maclaurin, best known for his
work on series, discusses spirals in Harmonia Mensurarum (1722).
We find parabolic spirals. In architecture there is the Ionic
capital on a column. In nature, the spiraled chambered nautilus is
associated with the Golden Ratio, which again is associated with the Fibonacci
Sequence.

Useful Links
and Books

http://wwwhistory.mcs.stand.ac.uk/history/Curves/Hyperbolic.html 
http://mathworld.wolfram.com/HyperbolicSpiral.html 
Boyer, Carl B., revised by U. C. Merzbach,
A History of Mathematics, 2nd ed., John Wiley and
Sons, 1991. 
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. 
Katz, Victor J., A History of Mathematics,
PEARSON
 Addison Wesley, 2004.

Lockwood, E. H., A Book of Curves,
Cambridge University Press, 1961. 
McQuarrie, Donald A., Mathematical
Methods for Scientists and Engineers, University Science Books, 2003.

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 animation contributed by
Gus Gordillo, 2004.


