Sascha Ledinsky wrote:

>> You did not explain what your macro does, and it is a bit hard
>> to follow without comments but here is what I think you are doing:
>> Lets call the corners A,B, and C. You split the triangle in three 
>> patches. One with corners: A, the middle of A and B, the middle
>> of A and C, and the centerpoint. Then you compute the remaining
>> points as functions of the originally given points. I can only
>> assume that the math is correct ;)
> 
> 
> This is a little off topic now, I think if we continue a discussion 
> about bezier patches we should start a new thread.
In a sense this is actually very on topic, it is on how one can use
such a technique in POV. But I agree that it is a separate subject
that crops up every now and then. Perhaps there should be a tutorial
on advanced bezier techniques, but I do not have the time to write
that :(

> Anyway, that's exactly what it does. I've found the algorithm in a 
> paper, I think it was:
> Hu Shi-Min.
> Conversion of a triangular Bezier surface into three rectangular Bezier 
> surfaces.
> Computer Aided Geometric Design, vol 13, 1996, 219-226.
> 
>> For POV to render these without cracks, you also have to
>> make sure that the division in triangles, as controlled by
>> u_step and v_step is the same in the patches in the triangle
>> and in the adjacent quad patches. This simply means that
>> in the triangle they have to be one lower then outside.
>> I think that it also forces a flatness of zero in the entire
>> set of patches. You have to be certain that the edges of
>> adjacent patches edges are split in the middle, so they
>> have to have flatness=0 and therefore their neighbours
>> also etc.
> 
> 
> You're right, but this is POV-Ray specific.
That is exactly why I posted it.
> I've only posted this to prove that it is possible to smoothly connect 
> three bezier patches to a triangular patch.
> My problem is that a cubic bezier triangle can't be connected smoothly 
> to cubic bezier patches 
Sorry, you lost me there. Exactly what is your problem?
(you may decide to go to povray.off-topic, I read that also)

> - what I'd need is a similar conversion 
> algorithm for something like Zheng-Ball patches (J.J. Zheng, A.A. Ball. 
> Control point surfaces over non-four-sided areas. Computer Aided 
> Geometric Design, 14:807-821, 1977) - discussed in this thread: 
>
http://news.povray.org/povray.binaries.images/thread/%3C3ff791bf%40news.povray.org%3E/?mtop=160788&moff=20

> 
> 
> All those papers have been available for free download on the internet 
> once, but now they have disappered and the only way to get them these 
> days is to pay $30 for each one at http://www.elsevier.com :-(
I am quite sure that I downloaded and printed that article then.
The paper will be in a stack somewhere and I probably have the PDF on
a backup disk.
There is a new elsevier policy that allows the authors to have a PDF
of their article for download at a page of their institution.

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