Interpolation on sphere with curve fitting toolbox

I'd like to know if it is possible to restirct the interpolation to the unit 3-sphere using curve fitting toolbox.

I have given points on that represent a not necessarily closed curve that i want to interpolate.

% generate points
n = 10;
a = linspace(0,2*pi,n);
r = 0.5;
v1 = sin(r).*cos(a);
v2 = sin(r).*sin(a);
v3 = zeros(1,length(a));
v4 = ones(1,length(a)).*cos(r);

% vectors / points on S^3
vec = [v1;v2;v3;v4];

% interpolation
curve_s = csapi(a,vec);

vecnorm(fnval(curve_s,[0:0.1:2*pi]))

The problem is that not all vectors that are reconstructed have .

Is there a way to restrict csapi that the reconstructed vectors have length 1 (are on the surface of the unit 3-sphere) ?

Thanks in advance

Answer 1

[Matt J]

You could pre-convert the input to spherical coordinates CART2SPH and then convert back after the interpolation.

[Drazen Sander]

The points that i have represent a closed curve in R^3. The reason i project them to the sphere in 4 dimesions is that i want to do some calculation in R^4.

So i can not use you solution because i have 4 coordinates. Anyway i also dont understand how this would solve the problem ? The points between the maped one that i get from the interpolation dont need to have length 1 by changing the representation of the point. Or do i miss something ?

[Matt J]

The reason i project them to the sphere in 4 dimesions is that i want to do some calculation in R^4.

This seems like a different question now from the one you posted: "I have given discrete points in R^3 that i map to S^3."

The points between the maped one that i get from the interpolation dont need to have length 1 by changing the representation of the point.

After converting to spherical coordinates, you would interpolate only the azimuthal and elevation coordinates, while leaving the radial coordinate unchanged.

[Drazen Sander]

Thank you. I edited my question i hope now it is more clear whats the problem.

[Matt J]

Similar to my original suggestion, you could convert your points to 4D spherical coordinates,

https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

and simply do the spline fit on the components.