I cannot access the paper, so I'm only guessing here to a large extent...
However, would piecewise $C^\{infty}$ not be OK to work with in terms of an
implementation? That is, if the contour had a fixed number of corners /
cusps that were known a-priori, could you not then apply said algorithm
between these points?
Just a guess though...
On 16/07/2013 2:00 PM, "phoenix" <[email protected]> wrote:
> Hi!
>
> After several weeks' work on the surface-surface intersections, it's
> improved significantly and can handle lots of cases that cannot be handled
> well before. It's more accurate and correct, ready for the evaluation.
> (Other intersections such as curve-curve, curve-surface, point-point,
> point-curve, and point-surface are also ready.) I'll still perform tests
> with some real geometries, such as the ones in db/*.g. If you are
> interested, you can go to the log and the images on my report page:
> http://brlcad.org/wiki/User:Phoenix/GSoc2013/Reports
>
> The major problem I encounter now is the overlapping cases, that is, two
> surfaces have some part overlapping, and we should report the boundary of
> the overlap region. As the current approach first computes intersection
> points, and then fit them into curves, so it can work well with transversal
> cases and tangent cases, but not overlap cases. It's quite hard to get the
> overlap region from these points. (For intersection points in the overlap
> region, the normals on them of the two surfaces should have the same or
> opposite direction, which can be used to determined whether they are inside
> an overlap region, and divide them into clusters according to their
> adjacency, forming the overlap region. But I think this is not very
> accurate, because these points are not necessarily on the boundaries.)
> Today I have read paper:
> http://www.sciencedirect.com/science/article/pii/S0010448596000991<http://www.sciencedirect.com/science/article/pii/S0010448596000991#>,
> and it mentions that given two polynomial surfaces, if they overlap over
> a region, the overlap region must be bounded by parts of the boundaries
> of the two surfaces. Although it specifies polynomial surfaces, but
> according to the prove, this seems to be satisfied for all continuous
> surfaces (C-infinite). If this property can be adopted, we can have a much
> simplified and accurate approach, because we can get the boundary of
> overlap directly.
>
> So my question is, can we assume that the surfaces are C-infinite? For the
> major primitives in BRL-CAD, this seems to be satisfied.
>
> Thank you and look forward to ideas and replies.
>
> Cheers!
> Wu
>
>
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