Polar
Equations
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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.
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The spiral on the tomb
of Jakob (James) Bernoulli.
Eadem mutata resurgo.
I shall arise the same though
changed. |
Applications
The Spiral of Cornu
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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.
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The trace of an underdamped
harmonic oscillator.
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| 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.
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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
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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 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.
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Useful
Links and Books
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| http://www-history.mcs.st-and.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.
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| Lockwood, E. H., A Book of
Curves, Cambridge University Press, 1961. |
McQuarrie, Donald A., Mathematical
Methods for Scientists and Engineers,
University Science Books, 2003.
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| 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.
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