# Visualizing a volume with MATLAB

I have a rational function $\phi: [0,1]^3 \to \mathbb R^3$ (A NURBS, to be precise) and I want to visualize the image of $\partial [0,1]^3$ using a surface plot (patch, for example). To do this I have chosen equidistant discretisation along each dimension and formed a meshgrid from this:

Nx = 50;
Ny = 2;
Nz = 20;

tx = linspace(0, 1, Nx);
ty = linspace(0, 1, Ny);
tz = linspace(0, 1, Nz);

[mx, my, mz] = meshgrid(tx,ty,tz);


Now I have implemented a function wich gives me a $3\times (Nx\cdot Ny\cdot Nz)$-matrix $M$ with the values $M_{:, i} = \phi(mx_i, my_i, mz_i)$ using linear indexing. Plotting this as a point cloud gives me a good gist of the figure: Now I'd rather like to have a "solid" plot of the shape by plotting a patch for each of the six sides of the deformed cube. Is there a nice way to do this without having to manually think through the indexing?

• If you have an equation, (or maybe even if not) you could try an isosurface. – David Feb 11 '15 at 4:41
• @David Thanks but the problem is I have a transformation while isosurface seems to be for a volume function. I've accomplished my needs with six surf plots above each other, one for each face of the cube. I'll post an answer later when home. – AlexR Feb 11 '15 at 8:10
• surf plots was going to be my next suggestion, I'll be interested to see how you handled it. – David Feb 11 '15 at 8:54
• @David I've posted my solution as an answer now. Feel free to post any improvements as an answer! I'll be sure to look at it. – AlexR Feb 11 '15 at 9:38

Finally, I've accomplished my goal by individually surf-plotting each face of the transformed cube:
After looking at the docs of surf I found sphere(n) to produce an input for surf which would render the unit sphere and looked into it: surf(X,Y,Z) where all are matrices of the same size produces the surface with these points preserving the rectangular topology. This leads to the following code:

[e1x, e1y] = meshgrid(tx,ty);
[e2x, e2z] = meshgrid(tx,tz);
[e3y, e3z] = meshgrid(ty,tz);
% These are the three different grids needed for the cube faces

e1x = e1x(:);
(...)
% Make all of them columns

e1z = ones(size(e1x));
(...)
% The respective third dimensions will be constant 0 or 1

bottom = [  e1x   e1y 0*e1z]';
top    = [  e1x   e1y 1*e1z]';
front  = [  e2x 0*e2y   e2z]';
back   = [  e2x 1*e2y   e2z]';
left   = [0*e3x   e3y   e3z]';
right  = [1*e3x   e3y   e3z]';
% Create the faces:
% surf(bottom(1,:), bottom(2,:), bottom(3,:)) with appropriate reshape
% would plot the bottom face of the unit cube

x = [bottom top front back left right];
Kstart = cumsum([1 size(bottom, 2) size(top, 2) size(front, 2) size(back, 2) size(left, 2)]);
Kend   = [Kstart(2:end)-1 size(x,2)];
% Concatenate and save the start- and end-indices of each face
Ndim = [Ny Nx;Ny Nx;Nz Nx; Nz Nx; Nz Ny; Nz Ny];
% Dimensions for each face


And then after evaluation of x into the variables Px, Py and Pz:

hold on
for k=1:length(Kstart)
Kcurr = Kstart(k):Kend(k);
% Index range of the points for the k-th face
surf(reshape(Px(Kcurr), Ndim(k,:)), ...
reshape(Py(Kcurr), Ndim(k,:)), ...
reshape(Pz(Kcurr), Ndim(k,:)), ...
k*ones(Ndim(k,:)), 'EdgeColor', 'none');
% surf each face assigning color k to face k for distinction
end


This produces the following image: Note that the red line is where two faces meet inside the figure.

• That's very nice, using Kstart like that was very clever! – David Feb 11 '15 at 21:36