[vtk-developers] Triangular patch interpolation

[Lin M]

Hi Dr. Thompson,

I have finished supplementary documentation for the vtkFiltersBezier
module. I think it's ready to be tested now.

On Thu, Aug 20, 2015 at 11:56 PM, Lin M <majcjc at gmail.com> wrote:

> Hi Dr. Thompson,
>
> I have added image based tests for almost all vtkFiltersBezier related
> class including BREP file reader, patch(simplicial) interpolation and point
> projection for both curve and surface example.
>
> I will complete required documents soon.
>
> Best,
> Lin
>
> On Tue, Aug 18, 2015 at 8:23 PM, David Thompson <
> david.thompson at kitware.com> wrote:
>
>> Hi Lin,
>>
>> It has been a while since we talked about that, and I am not sure from
>> looking at the method whether Pt is supposed to be the input or output. If
>> the input, then it should be named something like coord instead of Pt. If
>> the output, then you are correct, it should be a "double*" and the
>> documentation should spell out how much memory it must point to. (Since in
>> general the n-th derivative is a symmetric tensor of rank n, there are
>> (n+1)(n+2)...(n+k-1)/(k-1) partial derivatives. This is related to Pascal's
>> simplex. See
>> https://en.wikipedia.org/wiki/Pascal%27s_pyramid#Parallels_to_Pascal.27s_triangle_and_Multinomial_Coefficients
>>  and
>> https://en.wikipedia.org/wiki/Differential_of_a_function#Higher-order_differentials
>>  .)
>>
>> It is fine to have different methods for curves, surfaces, and volumes
>> for now. As you can see in the links above, the general pattern is that the
>> n-th total derivative is a tensor of all possible combinations of partial
>> derivatives with respect to the 1, 2, or 3 parametric coordinates. Call the
>> parametric coordinates r_i for i in {1, 2, 3} and say we are taking the
>> n-th derivative of f. Then Python code to compute one possible ordering of
>> the partial differentials in the output array would be
>>
>> def pdiff(i,n):
>>     r=[]
>>     if i == 1:
>>         return [[n,],]
>>     for x in range(n+1):
>>         for y in pdiff(i-1,n-x):
>>             r.append(y+[x,])
>>     return r
>>
>> which returns a list of i-tuples that all sum to n. For example
>>
>> pdiff(2,1) returns [[1, 0], [0, 1]]
>> pdiff(2,2) returns [[2, 0], [1, 1], [0, 2]]
>>
>> pdiff(3,1) returns [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>> pdiff(3,2) returns [[2, 0, 0], [1, 1, 0], [0, 2, 0], [1, 0, 1], [0, 1,
>> 1], [0, 0, 2]]
>>
>> However, really only the first (Jacobian, a vector) and second (Hessian,
>> a symmetric 3x3 matrix with 6 unique entries) derivatives are used very
>> much because (a) the number of terms grows very fast with increasing
>> derivative and (b) the first 2 derivatives are used widely for root-finding
>> and curvature computation but further derivatives are not.
>>
>>     David
>>
>> On Aug 17, 2015, at 22:40, Lin M <majcjc at gmail.com> wrote:
>>
>> Hi Dr. Thompson,
>>
>> I found you left an interface for
>> vtkBezierPatchAdaptro::EvaluateDeriv(double Pt[3], double* Paras, int d).
>> For 1-d curve, the derivative is always a 3*1 vector, but with the
>> parametric dimension increasing, higher order derivatives will contain
>> many points (it becomes matrix-by-vector differential). I can't figure out
>> a generalized way to interpret this. That's why I declared functions to
>> compute derivative for curve, surface and volume separately. Do you have
>> any suggestions for that? Thanks!
>>
>> Best,
>> Lin
>>
>> On Tue, Aug 18, 2015 at 1:04 AM, David Thompson <
>> david.thompson at kitware.com> wrote:
>>
>>> Hi Lin,
>>>
>>> > I have some questions about the test.
>>> > 1. About simplicial interpolation. Should I test some simple cases
>>> (hard coded simple shape) or some shapes like the motorcycle?
>>>
>>> Those are 2 different types of tests. (There are many; see
>>> http://stackoverflow.com/questions/437897/what-are-unit-testing-and-integration-testing-and-what-other-types-of-testing-s
>>> for some lists of test types). The first test (simple hard-coded shape) is
>>> a unit test and the second (test of both the reader, spline-to-patch class,
>>> and patch interpolation) is an integration test.
>>>
>>> Both are important, but since you already have an integration test for
>>> the motorcycle, I think a good set of unit tests for simplicial
>>> interpolation would be enough.
>>>
>>> > If it is the latter one, how to convert the rectangular patch to
>>> simpicial patch?
>>> >
>>> > 2. About point inversion. How to test it for the simpicial patch? I
>>> only implemented the point inversion for NURBS patch.
>>>
>>> If you didn't implement it, you can't test it. :-) However, if you know
>>> what you want the test to look like, you can write the test before you
>>> implement inversion and just leave it commented out with a note. Many
>>> people think this is how development should be done in the first place,
>>> since it forces you to consider how people will use what you write.
>>>
>>> > 3. About derivative computation. Should I just test them numerically?
>>> For example, given a shape and a parametric coordinate, test the result of
>>> our implementation with ground truth.
>>>
>>> Again, since you are using the derivatives in code that is part of an
>>> integration, I think unit testing (as you suggest) is enough.
>>>
>>>         David
>>
>>
>>
>
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