Supercurves: Superparabola and Superellipse

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NCB Deposit # 148

Dr. Cye Waldman
cye@att.net

Supercurves

Superparabola and Superellipse

in the Method of Archimedes . . . .

Applications from the Cylinder Hoof Created in MATLAB

Hoof in blue, Complement in red.

In MATLAB . . . . . . .

General Hoof Animation

Superparabola A sampler of curves in the superparabola family. Note the pulse for large values of p. We define superparabola as a parabola raised to a positive power greater than or equal to zero. Example:	Superellipse A sampler of curves in the superellipse family. Note the cusp for small values of p. The superellipse, also known as the Lamé curve, is used in the following form for the cylinder footprint:

Hoof, Complement & Cylinder Volume vs. Parameter p.	Hoof Volume vs. Parameter p.

Propositions 13 and 14 of The Method.

From prism to cylinder to half-cylinder to triangular slices of area formed on the "hoof."

animationprop14

To see any of the image full size, place the cursor over the image, right click and select "View Image."

Copyright Notice: The animations and all images within are under copyright by Cye Waldman and may not be copied, electronically or otherise, without his express permission.

Dr. Cye Waldman, cye@att.net

References

The half-cylinder, hoof and complement
illustrated with 3-D Printer models.

This web site accompanies the following journal publication:
Gray, Shirley B., Daniel Ye Ding, Gustavo Gordillo, Samuel Landsberger and Cye Waldman; "The Method of Aarchimedes: Propositions 13 and 14," Notices of the American Mathematical Society, vol. 62 (October, 2015).

Secondary Sources

J. L. Heiberg, Eine Neue Archimedeshandschrift, Hermes, Volume 42, 1907, pp. 235-303.
J. L. Heiberg and H. G. Zeuthen, Eine Neuve Schrift des Archimedes, Verlag von B. G. Teubner, Leipzig, 1907, pp. 321-363.
Thomas L. Heath, The Method of Archimedes: Recently Discovered by Heiberg, Cambridge, 1912.

Other Waldman contributions to the NCB using MATLAB:
More Valentine/Hearts:   < ..//waldman8/waldman8.htm >
Sinusoidal Spirals < ..//waldman/waldman.htm >
Bessel Functions    < ..//waldman2/waldman2.htm >
Gamma Funcions   < ..//waldman3/waldman3.htm >
Polynomial Spirals and Beyond   < ..//waldman4/waldman4.htm >
Fibonacci and Binet Spirals with a touch of Mondrian < ..//waldman6/waldman6.htm >
"Other" Fibonacci Spirals and Binet Spirals < ..//waldman7/waldman7.htm >
Voderberg Tiling   < ..//waldman9/waldman9.htm >
Cornu-Voderberg Tilings < ..//waldman10/waldman10.htm >
Aleph Animation < ..//waldman11/waldman11.htm >

2015