Asked by Markus Toivonen
on 4 Jul 2018

I have measured electric field data in three dimensions of the following form:

pos = [x y z]

ef = [e_x e_y e_z]

The matrices are 1000x3 in size, and the positions are located in a half sphere (cartesian coordinates). I want to be able to interpolate the electric field at some point in space, so that I receive all the three values of the electric field components, not just the norm. interp3 won't work as the points are not in a grid. scatteredInterpolant needs the norm as the input, so it does not fit my needs. Any suggestions on what function to use?

Answer by Anton Semechko
on 4 Jul 2018

Hi, Markus, here is a demo of how to perform linear interpolation of vector fields on a unit half-sphere:

half_sphere_interpolation_demo

function half_sphere_interpolation_demo

% Demo of how to perform linear interpolation of scalar and vector fields

% defined on a unit half-sphere.

% PART 1: Simulate data sampled on a half-sphere

% =========================================================================

% Generate a set of N randomly distributed points on the northern hemisphere

N=1E3;

X=RandSplHalfSphere(N);

% Triangulate points

x=StereoProj(X);

Tri=delaunay(x);

TR=triangulation(Tri,X);

% Generate vector field sampled at X

t=X(:,3);

t(t>1)=1;

t=acos(t);

f=@(t) sin([4*t 8*t 12*t]);

F=f(t);

figure('color','w')

TTL={'Fx' 'Fy' 'Fz'};

CLim=zeros(3,2);

for i=1:3

ha=subtightplot(2,3,i,[0.1 0.05],[0.05 0.05]);

h=trimesh(TR);

set(h,'EdgeColor','k','EdgeAlpha',0.25,'FaceColor','interp','FaceVertexCData',F(:,i))

axis equal

colorbar(ha,'Location','SouthOutside')

set(get(ha,'Title'),'String',TTL{i},'FontSize',20)

CLim(i,:)=get(ha,'CLim');

end

drawnow

% PART 2: Linear interpolation

% =========================================================================

% Generate new set of M points on a half-sphere

M=1E4;

Y=RandSplHalfSphere(M);

% Find faces of TR and containing Y

[T,uvw]=SphericalTriangleIntersection(TR,Y);

id_val=~isnan(T); % points in Y that intersect with TR

% IMPORTANT: In this demo, TR does not completely cover the entire

% northern hemisphere, and thus it will not be possible to

% interpolate F at some of the points in Y (close

% to the equator) that do not intersect with TR. Indices

% corresponding to these points are isnan(T).

% Only retain points in Y that intersect with TR

Y=Y(id_val,:);

T=T(id_val,:);

uvw=uvw(id_val,:);

% Interpolate F at Y

T=Tri(T,:); % triangles containing Y

F_int=bsxfun(@times,uvw(:,1),F(T(:,1),:)) + ...

bsxfun(@times,uvw(:,2),F(T(:,2),:)) + ...

bsxfun(@times,uvw(:,3),F(T(:,3),:));

% Visualize interpolation errors for each component of F

t=Y(:,3);

t(t>1)=1;

t=acos(t);

F_ref=f(t);

dF=F_int-F_ref;

TR2=triangulation(delaunay(StereoProj(Y)),Y);

TTL={'Fx error' 'Fy error' 'Fz error'};

for i=1:3

ha=subtightplot(2,3,3+i,[0.1 0.05],[0.05 0.05]);

h=trimesh(TR2);

set(h,'EdgeColor','k','EdgeAlpha',0.25,'FaceColor','interp','FaceVertexCData',dF(:,i))

axis equal

colorbar(ha,'Location','SouthOutside')

set(get(ha,'Title'),'String',TTL{i},'FontSize',20)

set(ha,'CLim',CLim(i,:))

colormap(ha,'jet')

end

function [T,uvw]=SphericalTriangleIntersection(TR,P)

% Find barycentric coordinates of the points of intersection between a

% hemispherical mesh, TR, and set of rays, P. Note that rays emanating from

% the origin = points on a unit sphere.

% Use stereographic projection (with north pole as the origin) to

% identify triangles intersected by the rays

x=StereoProj(TR.Points);

p=StereoProj(P);

tr=triangulation(TR.ConnectivityList,x);

[T,uvw]=pointLocation(tr,p);

function x=StereoProj(X)

% Stereographic projection from a unit sphere onto a plane tangent to the

% north pole.

x=bsxfun(@rdivide,X(:,1:2),1+X(:,3));

function X=RandSplHalfSphere(N)

% Use rejection sampling to generate N random points on the northern

% hemisphere

X=[];

while size(X,1)<N

Xi=randn(N,3);

Xi(Xi(:,3)<0,:)=[];

X=cat(1,X,Xi);

end

X=X(1:N,:);

X=bsxfun(@rdivide,X,sqrt(sum(X.^2,2)));

% -----------------------------------------------------------------

% -----------------------------------------------------------------

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## Aarti Dwivedi (view profile)

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