Re: [CAD_CAM_EDM_DRO] digitized point smoothing algorithm

Dear Chris, Buck, Hoyt and others,

Thanks for your thoughtful answer. I am taking all these ideas in. I do
not want to use any approach which requires a human to make decisions from
one digitize run to another, so, I might be forced to settle for some point
smoothing rather than curve fitting. Several approaches are interesting,
and I'm looking them all over.

I was advised by my Prof. friend at UMR to get a text book on analytic
geometry, which I plan to do. Calculus is over my head at this point.

Thank you all for your help

Tom Eldredge

----- Original Message -----
From: Chris Stratton <stratton@...>
To: <CAD_CAM_EDM_DRO@yahoogroups.com>
Sent: Wednesday, June 13, 2001 10:46 AM
Subject: Re: [CAD_CAM_EDM_DRO] digitized point smoothing algorithm


> > I realize this is a big general question, but I have seen bigger
> > ones on this forum, and I thought I would ask and see what happens.
> > It seems like Chris Stratton, the instrument maker might be able to
> > help, if I am able to follow his explanation with my limited mental
> > faculties.
>
> Oh no, what have I gotten myself into?
>
> Curve fitting is tricky, and I'm not very good at it.
>
> But basically, it can work like this:
>
> 1) Guess a general form of an equation that might fit your data. It
> used to be important to do this 'right' however with todays' computers
> we can take advantage of the fact that higher math shows that anything
> can be approximated (often inefficiently) as a polynomial, like
>
> y = Ax^4 + Bx^3 + Cx^2 + Dx^1 +E
>
> This is a 4th-degree polynomial as the highest power of X is the 4th,
> and many shapes may require higher degree ones, but the basic idea is
> the same. What the computer does for us is find values of the
> coefficients A, B, C, D, E, etc
>
> 2) Define a test that will tell us how well the curve with certain
> proposed values of the coefficients A, B, C, etc fits the actual data.
> For example, you might at each point calculate the square of the
> difference between the measued value and the one given by the trial
> equation. The sum of all these squared differences will give you an
> idea of how well your curve fits.
>
> 3) Now you try a bunch of possibilities for the coefficients and see
> which gives you the lowest error. There are no doubt tricks for doing
> this wisely, but you can also try to do it by brute force with a
> nested approach like this:
>
> For A = -limit to +limit
> For B = -limit to +limit
> For C = -limit to +limit
> For D = -limit to +limit
> test fit
> save values if it is lowest error yet seen
> next D
> next C
> next B
> next A
>
> Beware this kind of search can get very big for a polynomial of large
> degree!
>
> ------------
>
> Microsoft Excell has, either natively or with certain add-ons, a
> tool for minimizing the value of a cell by changing values of other
> cells. You can set up a spreadsheet with your measured data,
> calculated approximation (use some other cells to store the
> coefficients), squared error, and finally its sum. Tell the optimizer
> to minimize the value of the error sum cell by varying the values of
> the coefficient cells.. Graph the results and see how you like them.
>
> I've sometimes multiplied the error in each cell by a scale factor,
> and manually adjusted those factors to make certain points (like the
> end points of the curve) more important so they come out exactly,
> while others are less exact.
>
> Some packages including matlab, the free gnuplot, etc have built in
> curve fitting. (finding the more recent versions of gnuplot can be
> tricky - I can't remember if I finally used it under windows or
> linux).
>
> Smoothing data is a different but related issue... You could do
> things like replace each point with the average of itself, the
> previous, and next points. Depending on what your data is doing,
> there are kind of weighted averages you might employ such as the
> geometric mean (square the values, sum, divide by 3, and square root).
>
> You can also simply edit out or adjust bad data by hand if your points
> are limited in number.
>
> One risk in polynomial approximations is that they can oscillate a bit
> rather than give smooth curves. So far I've been lucky, but I'm
> generally fitting fairly simple shapes (diameter vs length for tapered
> and ultimately flared french horn parts).
>
> You can test for oscillation by repeatedly taking the derivative and
> looking for zero crossings in it. Taking the derivative of a
> polynomial is simple: write each term as A x^n and replace it with nA
> x^(n-1). You'll end up with one fewer term than you started with, as
> n is zero for the constant term.
>
> I'm sure others will have things to say, and hope I'll learn something
> from their comments.
>
> Chris
>
> --
> Christopher C. Stratton, stratton@...
> Instrument Maker, Horn Player & Engineer
> 22 Adrian Street, Somerville, MA 02143
> http://www.mdc.net/~stratton
> NEW PHONE NUMBER: (617) 628-1062 home, 253-2606 MIT
>
>
>
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