[MATLAB]: How would I mathematically and visually reproduce the 3D surface of the new King's Cross 'Western Concourse'?

Question

Anyone have any starting tips for me? I want to learn from this (ie Don't want to be lazy and have someone answer this for me).

I would like to develop my understanding of mathematical 3D surfaces. My own personal project is to produce a 3D surface/graph of the concourse structure in MATLAB.

I found a link with good pictures of its geometry here. I am not expecting to get it 100% perfectly but I'd like to come close!

At the end of this exercise I would like to have a mathematical definition of the geometry as well as a visual representation of the surface. This can involve cartesian equations, parametric equations, matrices, etc.

Any help would be very much appreciated!

Answer 1

To give some specific advice for MATLAB:

I would load in the 'section' image from the web page you have linked, and display this in a MATLAB figure window. You can then try plotting lines over the top until you find one that fits nicely. So you might do something like:

A = imread('~/Desktop/1314019872-1244-n364-1000x707.jpg');
imshow(A)
hold on
axis on

%# my guess at the function - obviously not a good fit
x = [550:900];
plot(x, 0.0001*x.^2 + 300)  

Of course, you might want to move the position of the origin or crop the picture and so on.

As an arguably better alternative to this trial-and-error method, you could trace the outline of the section (e.g by clicking points with something like ginput), and then use one of MATLAB's curve-fitting tools (e.g. fit) to fit a function to the data.

The final 3D shape looks to me (at a casual glance) to be a 3D revolution of the section shape around a central axis. Use of a cylindrical coordinate system could therefore be a good idea.

The final plotting of your 3D shape could be done with a function such as surf or mesh.

Answer 2

I would start by defining a function that defines for each x, y coordinate whether there is a point z, and if so with which altitude.

The shape reminds me a bit of a log or a square root.