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 NCB Deposit  # 117

Dr. Cye Waldman
cye@att.net


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Sinusoidal Spirals  

"The Apple of My i"


Cauchy
Cauchy
1789-1857

Waldman investigates spirals graphed in the imaginary plane .

(Click here for the full Article including Equations and Matlab Code)
sinus curve apple

"The Apple of My  i "

 
Additional examples of curves generated by the Cauchy pulse are shown below.
Lattice π
We obtain a curve somewhat like the limaçon if we conside the π th derivative of the Cauchy pulse (n = π).

Lattice
The image appears to consist of seven overlapping leaves though there are very few in nature.

The Centurion
If we allow real umber n-values to increase, we obtain increasing number of nested loops.  This animation is for n = 100.

Centurion



References, Comments and Matlab Code

Matlab Code for  "The Apple of My  i"  can be expressed in only four lines:

psi=inline('1./(1-i*tau).^(n+1)','tau','n');
tau=linspace(-1000,1000,1e6+1)';
n=i;
figure; plot(psi(tau,n)); axis equal

Or, in terms of the angular variable:

psi=inline('cos(theta).^(n+1).*exp(-i*(n+1)*theta)','theta','n');
theta=linspace(-pi/2,pi/2,1e6+1)';
n=i;
figure; plot(psi(theta,n)); axis equal

Hahn, Stefan,  Hilbert Transforms in Signal Processing,  Artech House, 1996.
Oliver, Frank, et al,    NIST Handbook of Mathematical Functions,  Cambridge, 2010.
Weisstein, Eric , CRC Encyclopedia of Mathematics,  CRC Press, Chapman and Hall, 2003.
Zwikker, C., The Advanced Geometry of Plane Curves and Their Applications, Dover Press, 1963.
Other Sinusoidal curves  < http://curvebank.calstatela.edu/spiral/spiral.htm >

  
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