Cross-sectional area of a cone(ish)

Morning all,

I have a surface that looks something like a lopsided squished cone, hollow in the middle.

I would like to take the cross-sectional area at different heights, but I need some help. I'm sure it shouldn't be too difficult but I can't figure it out today. I can use the countour function to plot the contours at different heights, but not find the area of the cross-section.

Can anyone help me please?

Answer 1

[John D'Errico]

And polyarea won't suffice to compute the area of a polygon? It should.

Just generate the contours as desired. Then use polyarea. Easy.

[jlt199]

Like this?

[C,H] = contour3(x,y,z,[zpk*0.5 zpk*0.5],'linewidth',2);
pkArea = polyarea(C(1,:),C(2,:));

[jlt199]

That works in about 90% of the cases, in noisy cases though I end up with this

Any suggestions on how to get around this? I'm only interested in the "cone"

[John D'Errico]

In that case, the contour is a noisy mess. You get what you get then. What did you expect? You asked for the area enclosed. If the surface is a noisy, bumpy mess, then the area enclosed will be difficult to measure, by ANY means. Computer algorithms are not magically gifted. They cannot read your mind, and the human eye is very good at seeing the signal the analyst wants to see, while removing the spurious dreck. (It would have helped if you stated originally this would be a noisy surface.)

1. Smooth the surface as necessary, BEFORE you compute the contours. This can introduce problems of course. Smoothing will change the surface. Too much smoothing may change it unacceptably. Too little and you get what you showed.

A simple scheme for smoothing a regular surface in 2 dimensions is to convolve the surface with a radially symmetric Gaussian blur kernel. conv2 will do this nicely, but you will need to choose the smoothing kernel. What will matter here is the standard deviation of the Gaussian.

Alternatively, you could use the equivalent of a Savitsky-Golay filter, which will implicitly fit a local 2-d surface at every point, then return only the predicted value of the surface at that point. Here the local polynomial order will matter, and the local span used for the fit. Again, this can be achieved using conv2, although I do not know of any implementations of the above scheme for 2-d smoothing. Not hard to write. :-P

I suppose you could also use my gridfit tool on the FEX, passing in the array as a list of (x,y,z) points in 3-dimensions. Gridfit allows you to control the smoothing parameter. Gridfit will be slower than other tools that are based on conv of course, so I'd first recommend one of the other schemes I mentioned.

There may be some 2-d smoothing tools on the FEX. If so, I cannot validate them for goodness of code. (Except for gridfit.)

Again, any smoothing scheme will do things like flatten and widen the peaks.

2. Break the contour polygons into their separate pieces, in case there are disjoint polygons. You will need to know which one is important to you then.

[John D'Errico]

Hmm. Just thinking that I could have offered an option in gridfit to allow choice of the smoothing parameter using a target RMSE for the surface. Surprising I did not. Oh well, something to add to the round tuit list.

[jlt199]

Thanks for your help John, I have just got back to this problem after having to leave it to write reports :(

I was as surprised by the level of noise as you, it's only in about 5 of about several hundred surfaces, so I missed it originally.

I'm having a hard time working out the structure of the matrix C which is output from the contour3 function, I have tried looking at the code for contour3, but really didn't understand what was going on. Can anyone help me with this? I would like to isolate polygons as suggested, but I can't understand the data structure.

Another thought I had was to just send the portion of the surface that contains the "horn" to the contour3 function, but I can't think how to do that either.

If anyone can help with either of these two problems I would be very grateful.

This is the surface I am currently working with:

Many thanks

[jlt199]

Ok, I've figured out the structure of C now. Now I just need to figure out what to do with it...

Answer 2

[Chad Greene]

I'd use my C2xyz function which is on File Exchange to easily get the x,y values of a contour line. Below I'm using the built-in peaks data as an example dataset and getting the area of the polygon bounded by the z=6 line.

[X,Y,Z] = peaks(1000); 

pcolor(X,Y,Z); 
shading interp
colorbar

zval = 6;
hold on
C = contour(X,Y,Z,zval*[1 1],'k');

[x,y,z] = C2xyz(C); 
A = polyarea(x{1},y{1})
A =
  0.6661

[Sean de Wolski]

How is C2xyz different than contourc?

[Chad Greene]

Hey Sean, the C2xyz function simply converts the difficult-to-interpret C matrix into more intuitive x and y values.

Answer 3

[jlt199]

Thanks, I've managed to get it working in most cases. Except where the noise gets horrendous.

Many thanks for your time