is a curve clockwise or counterclockwise?

[Nick Trefethen]

Anne Greenbaum has asked if Chebfun has a simple way to check if a curve is oriented clockwise or counter-clockwise. Here's one idea, but perhaps others can improve on it?

At the example http://www.chebfun.org/examples/geom/Area.html by Stefan Guettel, one sees a way to compute the area enclosed by a curve.  One could say that if the area is positive, then the curve runs counterclockwise, like this:

```

>> t = chebfun('t'); z = exp(pi*1i*t);

>> f = exp(z); area_of_f = sum(real(f).*diff(imag(f)))

area_of_f = 4.997133057057813

```

If the area is negative, then the curve runs clockwise, like this:

```

g = exp(1./z); area_of_g = sum(real(g).*diff(imag(g)))

area_of_g = -4.997133057057813

```

Do people have any suggestions about these ideas?

[Anthony Austin]

This was exactly the sort of thing that popped into my mind when I saw
this question, though I probably would have phrased it as integrating
1/(z - c) over the curve, where c is some point that lies within the
curve, and seeing if the result comes out positive or negative.

The problem is how to choose c. For convex curves, you can take c to be
the centroid, which you can easily compute by doing another Chebfun
integral. This is probably good enough for Prof. Greenbaum's
application, since I'm guessing that her curves are boundaries of fields
of values of matrices, and those sets are always convex. For non-convex
curves, you'd have to choose some other point.

At any rate, the approach based on Dr. Güttel's method doesn't have this
problem, and it saves an integral, so you may as well use it instead.


Regards,

Anthony P. Austin
aus...@maths.ox.ac.uk

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[Anne Greenbaum]

Thanks!  I will give the Guettel idea a try.  The curves that I am looking at 

are convex, but I would like to write a code that works for general closed curves.

Anne

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[Jacob Rus]

For a complex curve, is there any obvious way to compute the contour
winding number relative to a point at infinity?

[Jacob Rus]

> For a complex curve, is there any obvious way to compute the contour
> winding number relative to a point at infinity?

(For instance, perhaps you could invert the function and then find the
winding number about the origin.)

[Jacob Rus]

Another possibility would be to integrate the signed curvature along
the whole curve.

[Anne Greenbaum]

Another method that seems to work is the following:

dL_arg=angle(diff(L));

dL_arg=unwrap(dL_arg-dL_arg(0));

direction = sum(diff(dL_arg))

If the curve is counterclockwise, direction should be +2pi, and if clockwise, direction should be -2pi.

Then you can just check the sign of direction.

Anne

On Mon, Aug 8, 2016 at 3:15 AM, Anthony Austin <aus...@maths.ox.ac.uk> wrote:

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