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[liudaisuda <>]

The function is defined as a function of x and y, for instance. x*x +y*y

On Wed, May 29, 2013 at 7:21 PM, 杨成林 [via cgal-discuss] <[hidden email]> wrote:

The triangle can be viewed as a 3D one. Every vertex of the triangle has a height which equals the density. The value you want is the volume between the triangle and the zero height plane.
The key is how your density is defined. Is it defined by a function or grid points or scattered points? Without this, it is hard to determine the best way.

在 2013-5-29 下午6:00，"liudaisuda" <[hidden email]>写道：

what do you mean by density is linear. Sorry for this naive question.

On Wed, May 29, 2013 at 5:26 PM, 杨成林 [via cgal-discuss] <[hidden email]> wrote:

It is better to asume the density is linear in a triangle, as Finite
Element Method does. This improves the approximation a lot though the
calculation of the area of a triangle is more complex.

2013/5/29, Tapadi <[hidden email]>:

>
> Here, you just found a failure case of the checking method we used so far
> here. Let's remember:
>
> 1. We assume the density function to be constant over each triangle of the
> polygon's discretization (that is an approximation)
> 2. To check the validity of this approximation, we compare the values at
> each triangle's vertices. If they are equal, we deduce that the function
> has
> good chances to be actually constant in the triangle.
>
> But this is not always true, your case is a good example. The function may
> have equal values at each vertice, but still have huge variations between
> those vertices. How to overcome this case? There are two straight
> solutions:
>
> a. Decide of an maximum area each triangle should have. If any triangle has
> an area greater than this max value, then subdivide it no matter what are
> the values at vertices. This max area should be chosen in function of the
> maximum frequency you can find in your function. We can discuss this point
> later if you want, for now just choose a max area value and decrease it
> until you get a good integral result.
>
> b. In addition to the comparison of function's values at each vertice, you
> can check for the function's partial derivatives at each vertice. If they
> are not near zero, subdivide the triangle. You can generalize this approach
> by checking for zero values of function's second, third derivatives, etc.
> Actually, this approach is considering the Taylor series of the function
> and
> checking for zero values till a given rank.
>
> Here you are!
> Best regards,
> Hugo Loi
> PhD student at Inria - Maverick team
>
>
>
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杨成林
Yang Chenglin

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