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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.


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 (1788-1827), 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

Spiral of Fermat


Lituus' Spiral


Sinusoidal Spiral



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 so-called 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
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.