MATLAB: Find tangent to set of 2D points

geometryinterpolationMATLABtangent

Hi All!

I have sets of 2D points, which form something like a circle, but NOT necessarily convex (see picture below). The points are given by two vectors (x and y coordinates), in clockwise order. Unfortunately, this is not a classic y=f(x) function, as there can be two y values to the same x. (Also there can be vertical lines).

I'd like to find a tangent vector for each point (preferably also in clockwise direction). Can you help me? I find none of MATLAB's interpolation functions useful, as they only work on functions, not on R->R^2 data.

I was thinking about using some sort of analytical solution, which uses e.g. 5 points, to calculate the tangent vector at the 3rd, or some sort of numerical differentiation method, but I couldn't find one. (I'm sure there has to be an algorithm out there for this…)

I think using interpolation for the whole set would be too much for this problem, but if nothing else works…

Best Answer

Your curve is somewhat noisy. I might think to point out that my interparc tool from the file exchange can be made to give estimates of the derivatives of the curve at any point along the curve. So it would return two numbers at a point that are essentially dx/dt and dy/dt. The ratio is the slope of the tangent line at that point.

The above works even for non-functional closed curves as you have, since interparc works in a parametric form. It computes a simple cumulative linear chordal arclength along the curve, then uses spline models x(t), y(t).

The problem is, your data is somewhat noisy, based on the curve we are shown. So that tangent line would be pointing all over the place.

Anyway, if you were to use interparc in that way, then it would be just as simple to do the work yourself. You do not provide your data. But, suppose you have data that is in the form of two vectors x and y, sorted in order along such a path in the (x,y) plane?

t = cumsum([0;diff(x(:)).^2 + diff(y(:)).^2)]);

So, t is the cumulative piecewise linear distance between each point, one to the next along your curve.

splx = spline(t,x);
sply = spline(t,y);

Differentiate each spline, then evaluate the ratio of the derivatives at the original points. That is the slope of the tangent line, at that point. If you have the curve fitting toolbox, you can use fnder to differentiate the splines.

tanslope = ppval(fnder(sply),t)./ppval(fnder(splx),t);

If no curvefitting toolbox, the derivative function is trivial to generate for the pp form that spline returns.

Again, the issue is that you may wish to produce a smoother estimate of the tangent slope. If you provide your data, and you need such a smoothed tangent line, just attach the data to a .mat file as a comment.

Related Solutions

MATLAB: Plot Natural Cubic Spline

You were close, in that csape can do it. (My SLM toolbox also has a natural cubic as an option, but there is no need to go that far.) The proper choice in csape is:

pp = csape(x,y,'variational');

The flag 'variational' comes from the calculus of variations, I assume. The calculus of variations can also be seen to call those corresponding end conditions the "natural" boundary conditions.

   'variational' : set end second derivatives equal to zero
                    (ignoring given end condition values if any)

A natural cubic spline is the one that has zero second derivatives at the ends.

As a check, does it produce what I expect? So here a function that has decidedly NON-zero second derivatives at the ends.

x = linspace(0,2*pi,10);
y = cos(x);
pp = csape(x,y,'variational');
pp2 = fnder(pp,2);
fnplt(pp2)
grid on

Yet as you see, the second derivatives at the ends are forced to zero. This is the classic example that I always used when teaching about splines, as to why a natural cubic spline might often be a poor choice.

So if we look at the fit that arises, compare the natural cubic to that produced by spline.

fnplt(pp,[0 1])
hold on
pps = spline(x,y);
fnplt(pps,[0 1])

In blue is the natural cubic, whereas the green curve is the result of spline, which uses not-a-knot end conditions, generally a safer choice. If we remember these curves are an approximation to cos(x), the green curve is clearly much better.

So while you can find a natural cubic in MATLAB, you need to look deeply enough, and know what you should be looking for. That is in general a good thing in this regard.

MATLAB: How to interpolate data using quartic or higher order polynomials? Similar to the cubic spline interpolator csapi

You don't really want to do so. Yes, you think you might. But you don't.

High order polynomials are easy enough to build, polyfit can generate them. Just use one order less than the number of data points. So two points exactly determine a line, 5 points a quartic polynomial, etc.

That is the good news. The bad news is high order polynomials will almost always cause you to be forced to post a new anguished question, like "Why does my high order polynomial interpolant do insane things?"

In fact, depending on your data, even something relatively low order like a cubic polynomial can cause problems. Those problems typically arise from two things. The first is poorly scaled data. So if your independent variable (I'll call it x) varies over the interval [0,1000], then cubing those numbers will result in values on the order of 1e15 for a 5th order polynomial. Now, when you add and subtract numbers that vary by 15 powers of 10, expect to have numerical problems in double precision arithmetic.

High order polynomials say 15 or so, almost always tend to have numerical problems, because now you are raising your numbers to seriously high powers. The linear algebra needed to compute those coefficients often starts to have problems now, because again, it works in double precision arithmetic. So unless you are very careful about issues like scaling and centering of your data, expect problems.

Next, polynomials always SEEM like a good idea. (Hey, I can think of a lot of things I've done that SEEMED like a good idea at the time.) After all, a Taylor series is just a polynomial, and they can represent almost anything. But the fact is, polynomials are squirrelly things. They squirm around, introducing lots of added extrema to data that never did anything bad.

The classic example is seen here:

x = -10:10;
y = 1./(1+x.^2);
p = polyfit(x,y,20);
Warning: Polynomial is badly conditioned. Add points with distinct X values, reduce the degree of the polynomial, or try centering and scaling as
described in HELP POLYFIT. 
> In polyfit at 75 
ezplot(@(x) polyval(p,x),[-10,10])
hold on
plot(x,y,'ro')

It turns out that the squiggles you see are not due to the warning message, although scaling the data would have allowed it to work without generating a warning message.

And NEVER use a polynomial to extrapolate. Ok, linear extrapolation tends to be moderately safe. Stop there though.

Set('soapboxmode','off')

Best Answer

Related Solutions

MATLAB: Plot Natural Cubic Spline

MATLAB: How to interpolate data using quartic or higher order polynomials? Similar to the cubic spline interpolator csapi

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