Gamma Function for Log-Aesthetic Curves

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NCB Deposit # 122

Dr. Cye Waldman
cye@att.net

"The Incomplete Gamma Function for Log-Aesthetic Curves"

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 . . . .

Archimedes
287 - 212 BC

Click here for the full article including Equations.

		Log Aesthetic Spiral Curves with parameter a = 4.
Our table shows the polynomial terms for the first few values of a.
"My work always tried to unite the true with the beautiful, but when I had to choose...I usually chose the beautiful."	"No mathematician can be a complete mathematician unless he is also something of a poet."	"It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, 'Yes, it is true, but you won't be able to understand the proof."
Hermann Weyl	Karl Weierstrass	Ronald Graham

Matlab Remarks

The calculations were carried out on a Pentium 6-core i7 3.20 GHz computer using Matlab. The exceptional speed of the DNS can be attributed to the Matlab function cumtrapz . This is in lieu of 10^5 - 10^6 for loop calculations.

Incomplete Gamma Spiral for Yin-Yang

incgamma

We confined our graphics to integer values of a.

References

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

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.

Other Waldman contributions to the NCB:
Sinusoidal Spirals: < ..//waldman/waldman.htm >
Bessel Functions < ..//waldman2/waldman2.htm >

Other spiral Deposits in the NCB:
< ..//spiral/spiral.htm >
< ..//log/log.htm >

2013