non uniform thickness calculations

Hi all; I have x,y,z coordinates and I need to get the thickness throughout length I have (straight and arch segments) as shown in the attachments. The problem the thickness is not uniform I do not know if anyone have any idea about that. Thanks

[IEngineer7Mof7]

Thanks for great ideas

Appreciated

Answer 1

[John D'Errico]

This will be true whenever you have two nonlinear relationships, describing the upper and lower envelopes. Even if you have two functions that are perfectly uniformly translated in y, the distance between those two functions at any point need not be constant, and will not in general be so.

What would you do with the "thickness"?

Consider two simple nonlinear relationships, over the interval [-1,1].

y = exp(x)
z = 1 + exp(x)

Clearly they are the same identical function, but the distance between the two functions at any point will not be 1. (So the distance here will be the distance between any two points of minimal Euclidean distance.)

We can solve for (and plot) that minimal distance in the above example.

y = @(x) exp(x);
z = @(x) 1 + exp(x);

As a function of x, find the point on curve z that is closest to y(x). It will not in general happen at x.

x = linspace(-1,1);
x_z = zeros(size(x));
mindist = zeros(size(x));
for i = 1:numel(x)
  [x_z(i),mindist(i)] = fminbnd(@(u) norm([x(i),y(x(i))] - [u,z(u)]),-3,3);
end

plot(x,x_z)
xlabel 'x'
ylabel 'location on z curve'

figure
plot(x,mindist)
xlabel 'x'
ylabel 'minimum distance between curves'

So the first figure is the location of closest approach on z(u) to y(x). The second figure is the actual distance computed. Note that the actual distance computed was always less than 1, which is the offset in y.

The point of all this is those curves are likely at a fairly constant offset in y, even though the thickness of that band does not appear to be uniform.

So the pertinent question is what would you do with the thickness? Does it really mean what you think it does, given what I have shown here? Is orthogonal distance truly pertinent?

[IEngineer7Mof7]

Thanks