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 NCB Deposit  # 149

Dr. Cye Waldman
cye@att.net
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Superconics

Superparabola, Superellipse, Hyperbola, Pseudo-Hyperbola and a
Generalized Superformula


superequation
Standard form of the Superconics


In MATLAB . . . . . . .
SuperconicAnimation
Superconics Animation

Code

Why Superconics?
Cye H. Waldman

There is a plethora of super this-and-that running around the mathematical world these days: superellipse (and its 3D cousin, superquadrics), supercircle, superparabola, superhyperbola, and superformula. So why do we need superconics?

Superconics is a system of curves, based on the conic sections, for two and three-dimensional shapes. With the exception of the superformula, all of the above are subsets of superconics. In one equation, superconics describes all of these and more. Moreover, it is amenable to exact mathematical analysis. Specifically, a single analytic solution gives the area, centroid, moments, and volume of bodies of revolution over the wide range of variables embraced by superconics.

Superconics provides a systematic set of functions for evaluation of various problems. This has allowed us to extract certain mathematical relations that may have gone unnoticed heretofore.

Superconics is essentially a two-parameter function, with one additional switch that distinguishes elliptic/parabolic and hyperbolic types. Extended superconics is a three-parameter function that provides the full spectrum of curves between and beyond elliptic and hyperbolic types. This function completely blurs the distinction between the two types; you can transition smoothly between them. The extended superconics likewise has a completely analytic solution.

As for the superformula, it’s merely an elaborate curve-fitting equation with four parameters in addition to axial scaling, cleverly guised in mathematical notation. There are no mathematical properties, per se, because it’s intractable. Its properties can only be determined by direct numerical simulation; but then it’s doubtful that any of its acolytes has ever looked.

In our studies of superconics we have come to develop new understandings that can be readily ported to other areas of interest. In the mathematical arena, we have demonstrated new relations between volume and centroid and the plane curves that generate them.

In our studies of superconics in three dimensions, we have developed an algorithm for 3D shapes that allow the generating curve (or vertical profile) to vary continuously while tracing out the base curve.

And that is why superconics.
cyesig



superellipse
                                        Superparabola
                           pulseparabola
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:

equationparabola

colorsuperellipse
                                        Superellipse
                              cuepellipse
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:

ellipse equation

Superconics:  Hyperspheroid

hyperspheroid2
and the equations . . . . .
table1


Another Version of a Superconics Hyperspheroid

hyperspheriod





Methods:  Writing the Program

All 3D figures and animations were created with a program developed by the authors. The crux of the program is the spherical product routine that is essentially the product of a matrix and a diagonal matrix. The input parameters are an n-by-n matrix (n-by-n column vectors for the vertical ribs) and an n-vector (the planiform curve).  These are as expressed complex variables; in the matrix, the real and imaginary parts are the radial and vertical components of the vertical ribs, respectively.

In the main program, the n-by-n matrix is calculated sequentially around the planiform based on the local vertical profile. All the vertical profiles are superconics, but the need not be. Toroids and Mobioids can be created with a few control parameter settings. And the same program applies to the extended superconics with 3 parameters or complex parameters.

We have used the same program to create 3D forms from regular and star polygons, cuspids and rosettes, Reuleaux triangles, superformulas, and more, even fractals (imagine a Koch snowball created form Koch snowflake). Moreover, any of these forms can be mixed and matched; the program does not care.

In addition, we have used the same program, albeit with some ad hoc modifications to create twisted cylinders, the Archimedean hoof, sphericon, tractroid, and more.

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

<  https://en.wikipedia.org/wiki/Superparabola  >.



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

 
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