S2VectorFieldHarmonic

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

Defining a S2VectorFieldHarmonic

Operations

Visualization

Complete Function list

Defining a S2VectorFieldHarmonic

Definition via function values

At first you need some vertices

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

Next you 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.approximation(nodes, y)

 
sVF1 = S2VectorFieldHarmonic  
 bandwidth: 128

Definition via function handle

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

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

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

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

 
sVF2 = S2VectorFieldHarmonic  
 bandwidth: 128

Definition via S2FunHarmonic

If you directly call the constructor with a multivariate S2FunHarmonic sF 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 and 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

addition/subtraction

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

multiplication/division

2.*sVF1; sVF1./4;

dot product

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

cross product

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

Other operations

mean value

mean(sVF1);

Rotation

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

pointwise norm of the vectors

norm(sVF1);

Visualization

the default plot-command

plot(sVF1);

same as quiver(sVF1)

3D plot of a sphere with the vectors on itself

clf;
quiver3(sVF2);