Re: Integrating scattered data

[kensmith]

On Tuesday 21 August 2007 08:46, Jordi Gutiérrez Hermoso wrote:
> On 21/08/07, kensmith <address@hidden> wrote:
> > On Monday 20 August 2007 09:27, Jordi Gutiérrez Hermoso wrote:
> > > I have a surface in some irregular domain of R^2 that I'm
> > > sampling at scattered, unstructured points. I'd like to find the
> > > volume under this surface.
> >
> > Do you know anything about how the surface gets from one place to
> > the next?
>
> What do you mean? No, I don't think I know that. They are scattered
> points without any structure.

To make my basic point clearer, let me give some examples of (X,Y,Z) 
data that I might gather so you can see what I mean.

Case 1:
I go to a local park taking my GPS, a not pad and a brick with me.  

I spin around three times and lob the brick up in the air.  It comes 
down somewhere.  I go there turn on my GPS and note down the (X,Y,Z) 
value for where the brick is resting.

The ground in the park follows smooth contours so if I gather enough 
data, I can estimate the volume of soil from mean sea level to the 
surface of the park.

Case 2:
I go to the same park also carrying a tape measure.  At each point, I 
measure how tall the person was that the brick hit.  I record zero if 
it didn't hit anyone.

The function of how tall people who are foolish enough to hang around 
when someone is lobbing a brick at random doesn't follow any sort of 
smooth contour.   Finding the total volume of foolish people with head 
injuries would be quite a different matter than Case 1.

>
> > Also, do you know what happens off the edges?
>
> I don't understand your question, but no, I don't think I have any
> special information about the surface around the edges.

Since, I assume you don't have measurements along the edges, you will 
have to assume something.

>
> > Things like Simpsons rule assume straight lines in how the curve
> > goes from point to point.  In effect, they are interpolating the
> > points between using that rule.
>
> I have heard of things like spline surfaces using tensor products or
> something like that. And Simpson's rule assumes parabolas between
> points, not straight lines. :-)

You are right.  There was a short circuit between the headphones.
>
> - Jordi G. H.

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