[vtk-developers] Class design for spline visualizations

[Lin M]

Hi Dr. Thompson,

Sorry for late reply. I have some questions here.

In the function:
void InterpolateOnPatch(vtkDataArray* outputPt, vtkDataArray* P_i, double*
r);

1. Do I assume that i and r are both a 3-vectors here? If not, how do I
determine the dimension of that?
2. Do I assume the highest degree of each polynomial dimension is equal to
each other, thus the maximum of (i,j,k) for P_{i,j,k} can be calculated by
max(i)=max(j)=max(k)=P_i->GetNumberOfTuples()/3?
3. Even I know the maximum of (i,j,k), what's the order of storage for the
P_i in memory?

Best,
Lin


On Fri, Apr 3, 2015 at 12:56 PM, David Thompson <david.thompson at kitware.com>
wrote:

> > My name is Lin. I'm interested in the project idea about spline
> visualizations from VTK/GSoC2015. I hope I can make some works on this
> topic and I understand that many preliminary discussions are needed for a
> new set of classes in VTK. This mail will contain the conversations about
> class design for this project.
>
> Hi Lin,
>
> One of the first things we should discuss is notation so we can be
> consistent and avoid confusion.
>
> Notation
> --------
>
> One of the first steps will be interpolating points on Bézier patches
> given control points, P_i, and parametric coordinates, r. Note that i and r
> may be 1-, 2-, or 3-vectors depending on whether the patch is a curve,
> surface, or volumetric region. Because VTK assumes points always have 3-D
> coordinates, each control point will always have 4 coordinates (3 spatial
> coordinates plus a denominator because we are interested in rational
> functions). So, a simple bilinear surface patch might be defined by
>
>   P_{0,0} = (0,0,0,1)
>   P_{1,0} = (1,0,2,1)
>   P_{1,1} = (1,1,0,1)
>   P_{0,1} = (0,1,1,1)
>
> and for r = (0.5, 0.5), P(r) = (0.5, 0.5, 0.75, 1). P(r) can be projected
> to a 3-D point by dividing the first 3 coordinates by the fourth. Let's
> call the projected version P'(r) = (0.5/1, 0.5/1, 0.75/1).
>
> As the degree increases, more control points will be required. The example
> above is for a rectangular patch. Bézier patches may also be triangular or
> tetrahedral by using barycentric coordinates.
> https://en.wikipedia.org/wiki/B%C3%A9zier_triangle
>
> VTK background
> --------------
>
> Because we are dealing with rational patches, one of the first issues will
> be how control points are represented. The vtkPoints class assumes that
> points always have 3 coordinates (not 4 like our control points). We can
>
> 1. Assume that control points are specified with 2 things: a vtkPoints
> instance plus a vtkDataArray holding the denominators.
> 2. Create a new vtkControlPoints class that inherits from vtkPoints and
> allows for an extra coordinate.
> 3. Not use vtkPoints instances at all and store control points in a
> vtkDataArray with 4 components per tuple.
>
> I am leaning toward option 2. Note that these classes do not force a
> particular memory layout because of recent changes to the VTK array
> classes. So, another related issue is how the interpolation techniques can
> be written to allow alternative memory layouts so that isogeometric finite
> element simulations can specify the memory layout instead of forcing the
> simulation to adopt VTK's.
>
> It would be a good idea to read this wiki page:
> http://www.vtk.org/Wiki/VTK/InSituDataStructures (especially the "Mapped
> vtkDataArrays" section) to understand how the array classes have changed
> recent to allow different memory layouts.
>
> A good first step would be to write a few tests to perform interpolation
> of some simple 1-, 2-, and 3-D patches given control points in a
> vtkDataArray and using the vtkDataArrayIteratorMacro to access control
> point coordinates. I can provide some control points, parametric
> coordinates, and the correct interpolated point values if you provide a
> function like
>
>   void InterpolateOnPatch(vtkDataArray* outputPt, vtkDataArray* P_i,
> double* r);
>
> that interpolates P_i(r) and stores the result in outputPt by calling
> outputPt->InsertNextTuple().
>
>         David
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