The Polynomial Spiral and Beyond

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NCB Deposit # 125

Dr. Cye Waldman
cye@att.net

"The Polynomial Spiral and Beyond"

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

Euler on a Swiss franc note.
1707 - 1783

A Polynomial Spiral is a generalization of the Spiral of Cornu, aka Euler's Spiral.
Click here for the full article including Equations.

One of the many ideas that led to the Spiral of Cornu was the desire to design auto and railroad track transitions that were free of "jerk," i.e., the derivative of the acceleration. In other words, we must avoid a sudden change in centrifugal force as would occur with a straight line to circular segment. Another application is in freeway design. A "straight" highway has zero curvature, but a gradual transition to the midpoint of the freeway exit must have nonzero curvature.

In addition to practical applications, spirals have a and lengthy fascinating heritage. Among others, Euler studied spirals in 1781 in connection with his investigations of elastic springs. His mentor, Jakob Bernoulli was also interested. Later, Marie Alfred Cornu (1841-1902), a French engineer and scientist, studied the curves in conjunction with the diffraction of light. (French usually do not use "Marie" for its confusion with a female name.) Many publications by others soon followed.

Waldman continues the exploration of log-aesthetic curves by an alternative approach. Quoting from his "pdf" file,

A new class of polynomial spirals?
He begins with the tangent angle formulation of the curve, . . .
alternative calculation

"Without apology or embarrassment, he (Euler) treated these numbers (real and imaginary) as equal players upon the mathematical stage and showed how to take their roots, logs, sines, and cosines."

"In mathematics you don't understand things. You just get used to them."

W. Dunham in Euler: The Master of Us All

Johann von Neumann

3-D Spirals

References

Burchard, H. G. et al. (1994). "Approximation with Aesthetic Constraints," in Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-aided Design, N. S. Sapidis, Ed., SIAM.

Dillen, F. (1990). "The Classification of Hypersurfaces of a Euclidean Space with Parallel Higher Order Fundamental Form," Mathematische Zeitschrift, 203: 635-643.

Gray, A., (1998). MODERN DIFFERENTIAL GEOMETRYof Curves and Surfaces with MATHEMATICA®, 2nd. ed., CRC Press.

Olver, F.W. J., Lozier, D.W., Boisvert, R.F., and Clark, C.W., (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.

Yoshida, N. and Saito, T. (2006). "Interactive Aesthetic Curve Segments," The Visual Computer, 22 (9), 896-905.

Ziatdinov, R., Yoshida, N., and Kim, T., (2012). Analytic parametric equations of log-aesthetic curves of incomplete gamma functions, Computer Aided Geometric Design, 29 (2), 129-140.

Zwikker, C. (1963). The Advanced Geometry of Plane Curves and Their Applications, Dover Press.

Other Waldman contributions to the NCB:
Sinusoidal Spirals: < ..//waldman/waldman.htm >
Bessel Functions < ..//waldman2/waldman2.htm >
Gamma Funcions < ..//waldman3/waldman3.htm >
Other spiral Deposits in the NCB:
< ..//spiral/spiral.htm >
< ..//log/log.htm >

from Jakob Bernoulli's tomb in the cathedral
at Basel, Switzerland.

Eadem mutata resurgo.
I shall arise the same though changed.

2013