Density flow of a vector field along the roman surface of Steiner
Prof. E.Musso, PhD.
Dipartimento di Matematica Pura ed Applicata
Università de L’Aquila- L’Aquila-Italy.
In this paper we consider the density flow of a vector field along the Steiner’s Roman surface. The components of the vector field are Wierstrass’s elliptic functions. Symbolic and numerical computations, as well the visualization, are performed with the software Mathematica 5.1.
Denisity flow of a vector field along the roman surface of Steiner
Fig. 1 : the roman surface of Steiner.
The roman surface is defined by the parametric equations
Fig. 2 : densità del flusso del campo vettoriale W
This picture represents the density of the flow of the vector field W with components
(4) W1(u,v) = Re[σ(u+iv, ω1+ iω2)],
(5) W2(u,v) = Im[P(u+iv, ω1+ iω2)],
(6) W3(u,v) = Im[σ(u+iv, ω1+ iω2)],
where P and σ denote the Weierstrass elliptic functions with fundamental periods ω1=1 and ω2=2. The density through the surface is given by
(7) ρ = N∙W (g11 g22 - g122)1/2,
where the functions gij , i,j=1,2, are the coefficients of the first fundamental form of the surface and N denotes the Gauss map. The visualization of the density is obtained by the means of the following “color function”
(8) CF(u,v) = Hue(-1/2(1+(1+ ρ)(1+ ρ2)-1),1,1)
The program has been written with the software Matematica 5.1.
Fig. 4: the program