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A plot resembling a ripples in water created with Clifford Legendre Polynomials



From the 2022 Maple Conference Art Gallery


 

 

Clifford Water Drop Wave

Hamed Baghal Ghaffari and University of Newcastle at Australia (UON)

Abstract

In this work we first write a peice of Maple code which returns the Clifford Legandre Polynomials. Then we use these polynomials to plot a water drop wave.

 

Keywords: Clifford Legendre Polynomials.

 

Preliminary Material

 

 

The even and odd Clifford legendre polynomials (CLPs) introduced in [1] can be calculated respectively as in below (see [2]).


C"`__2N,m`(`Y__k`)(x)=(2^(2 N)(2 N)!)/(N!)(∑)((N) ? (l))(GAMMA(l+k+(m)/(2)+N))/(GAMMA(l+k+(m)/(2)))(-1)^(l)|x|^(2 l)`Y__k`(x),"NULL 

C"`__2N+1,m`(`Y__k`)(x)=(2^(2 N+1)(2 N+1)!)/(N!)(∑)((N) ? (l))(GAMMA(l+k+(m)/(2)+N+1))/(GAMMA(l+k+(m)/(2)+1))(-1)^(l)|x|^(2 l)x`Y__k`(x),"

where m is the dimension, Y
"`__k` is a homogenous monogenic polynomial, and N is the order of the polynomial. We plot the radial part of the CLPs in dimension 2."

Maple Code
 

Even Clifford Legandre Polynomials.

 

evencliffordlegendrewithoutyk := proc(r, N, k)

proc (r, N, k) local y, i, C; if N = 0 then C := r^k/sqrt(2*Pi) else C := 2^(2*N)*GAMMA(2*N+1)*GAMMA(k+N+1)*r^k/(GAMMA(N+1)*GAMMA(k+1)*sqrt(2*Pi)); for i to N do C := C+2^(2*N)*GAMMA(2*N+1)*(-1)^i*binomial(N, i)*GAMMA(i+k+N+1)*r^(2*i+k)/(GAMMA(N+1)*GAMMA(k+i+1)*sqrt(2*Pi)) end do end if; y := C*sqrt(2*k+4*N+2)/(2^(2*N)*factorial(2*N)); return y end proc

Plots

 

Here we show some initial plots. For the main art work, see the end of the file.

 

> 

plot3d([evencliffordlegendrewithoutyk(sqrt(x^2+y^2), 1, 1)], x = -1 .. 1, y = -1 .. 1)

 

 

 

It is easy to play a bit with the parameters to ger a wave-like picture.

 

> 

p1 := plot3d([evencliffordlegendrewithoutyk(sqrt(x^2+y^2), 3, 2)], x = -1 .. 1, y = -1 .. 1, color = cyan); plots:-display(p1, view = [-.65 .. .65, -.65 .. .65, -2 .. 2])

 

 

Now we will plot a nice Clifford water drop wave.

 

> 

Hamed := piecewise( x^2 + y^2 < 10, 1/10*evencliffordlegendrewithoutyk(sqrt((x/3)^2 + (y/3)^2), 10, 1), -10 ):

> 

p := plot3d([Hamed], x = -5 .. 5, y = -5 .. 5, color = blue, style = surface, grid = [200, 200]):
plots:-display(p, view = [-2 .. 2, -2 .. 2, -1 .. 1],axes = none);

 

 

Odd Clifford Legandre Polynomials.

 

oddcliffordlegendrewithoutyk := proc(r, N, k)

proc (r, N, k) local y, i, C; if N = 0 then C := -2*r^(k+1)/sqrt(2*Pi) else C := -2^(2*N+1)*factorial(2*N+1)*GAMMA(k+N+2)*r^(k+1)/(factorial(N)*GAMMA(k+2)*sqrt(2*Pi)); for i to N do C := C-2^(2*N+1)*factorial(2*N+1)*(-1)^i*binomial(N, i)*GAMMA(i+k+N+2)*r^(2*i+k+1)/(factorial(N)*GAMMA(k+i+2)*sqrt(2*Pi)) end do end if; y := C*sqrt(2*k+4*N+4)/(2^(2*N+1)*factorial(2*N+1)); return y end proc

 

And just some more plots with the odd Clifford Legandre polynomials for fun.

 

> 

plot3d([oddcliffordlegendrewithoutyk(sqrt(x^2+y^2), 1, 1)], x = -1 .. 1, y = -1 .. 1, color = cyan)

 
> 

p2 := plot3d([oddcliffordlegendrewithoutyk(sqrt(x^2+y^2), 2, 1)], x = -1 .. 1, y = -1 .. 1); plots:-display(p2, view = [-.65 .. .65, -.65 .. .65, -2 .. 2], color = "Moccasin")

 

Acknowledgements

 

The author is supported by the Australian Research Council through Discovery Grant DP160101537 and Lift-off fellowship from Australian Mathematical Society.

References

 
1. 

Delanghe, R., Sommen, F., & Soucek, V. (2012). Clifford algebra and spinor-valued functions: a function theory for the Dirac operator (Vol. 53). Springer Science & Business Media, https://doi.org/10.1007/978-94-011-2922-0
2. 

Ghaffari, H. B., Hogan, J. A., & Lakey, J. D. (2022). Properties of Clifford-Legendre Polynomials. Advances in Applied Clifford Algebras, 32(1), 1-25, https://doi.org/10.1007/s00006-021-01179-8

 


 

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