S2VectorField

S2VectorField handles three-dimensional functions on the sphere. For instance the gradient of an univariate S2FunHarmonic can return a S2VectorFieldHarmonic.

Defining a S2VectorFieldHarmonic

Definition via function values

At first we need some example vertices

nodes = equispacedS2Grid('points', 1e5);
nodes = nodes(:);

Next, we define function values for the vertices

y = vector3d.byPolar(sin(3*nodes.theta), nodes.rho+pi/2);

Now the actual command to get sVF1 of type S2VectorFieldHarmonic

sVF1 = S2VectorFieldHarmonic.interpolate(nodes, y)

sVF1 = S2VectorFieldHarmonic
 bandwidth: 224

Definition via function handle

If we have a function handle for the function we could create a S2VectorFieldHarmonic via quadrature. At first lets define a function handle which takes vector3d as an argument and returns again vector3d:

f = @(v) vector3d(v.x, v.y, 0*v.x);

Now we can call the quadrature command to get sVF2 of type S2VectorFieldHarmonic

sVF2 = S2VectorFieldHarmonic(@(v) f(v))
% sVF2 = S2VectorFieldHarmonic.quadrature(@(v) f(v))

sVF2 = S2VectorFieldHarmonic
 bandwidth: 128

Definition via S2FunHarmonic

If we directly call the constructor with a multivariate S2FunHarmonic with two or three entries it will create a S2VectorFieldHarmonic with sF(1) the polar angle and sF(2) the azimuth or sF(1), sF(2), and sF(3) the \(x\), \(y\), and \(z\) component.

sF = S2FunHarmonic(rand(10, 2));
sVF3 = S2VectorFieldHarmonic(sF)

sF = S2FunHarmonic(rand(10, 3));
sVF4 = S2VectorFieldHarmonic(sF)

sVF3 = S2VectorFieldHarmonic
 bandwidth: 3
 
sVF4 = S2VectorFieldHarmonic
 bandwidth: 3

Operations

Basic arithmetic operations

Again the basic mathematical operations are supported:

addition/subtraction of a vector field and a vector or addition/subtraction of two vector fields

sVF1+sVF2; sVF1+vector3d(1, 0, 0);
sVF1-sVF2; sVF2-vector3d(sqrt(2)/2, sqrt(2)/2, 0);

multiplication/division by a scalar or a S2Fun

2.*sVF1; sVF1./4;
S2Fun.smiley .* sVF1;

Warning: M is too large for useing an precomputed
optimal quadrature grid. Instead we use
GaussLegendre quadrature.

dot product with a vector or another vector field

dot(sVF1, sVF2); dot(sVF1, vector3d(0, 0, 1));

cross product with a vector or another vector field

cross(sVF1, sVF2); cross(sVF1, vector3d(0, 0, 1));

mean vector of the vector field

mean(sVF1);

rotation of the vector field

r = rotation.byEuler( [pi/4 0 0]);
rotate(sVF1, r);

pointwise norm of the vectors

norm(sVF1);

Visualization

One can use the default plot-command

plot(sVF1);

or the 3D plot of a sphere with the vectors on itself

clf;
quiver3(sVF2);