POV-Ray : Newsgroups : povray.general : Curved Triangles Server Time
7 Dec 2021 09:56:35 EST (-0500)
  Curved Triangles (Message 1 to 6 of 6)  
From: Ben Chambers
Subject: Curved Triangles
Date: 27 Jun 2002 03:17:20
Message: <3d1abc00@news.povray.org>
Once again, folks, I'm at the curved triangle thingy!!!

Anyway, I think this attempt came out better.  Take a look at my picture in
p.b.i and, if interested, the source in p.b.s-f.  The include file contains
a description of how to use it, but if you have any questions feel free to
ask.

BTW, it's late here, and I haven't had time to try any meshes with it.
Please let me know if you run into problems doing so (smooth triangles
should convert without problems, but you never know).

...Chambers


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From: John VanSickle
Subject: Re: Curved Triangles
Date: 27 Jun 2002 14:30:47
Message: <3D1B59E3.54F30927@hotmail.com>
Ben Chambers wrote:
> 
> Once again, folks, I'm at the curved triangle thingy!!!
> 
> Anyway, I think this attempt came out better.  Take a look at my
> picture in p.b.i and, if interested, the source in p.b.s-f.  The
> include file contains a description of how to use it, but if you
> have any questions feel free to ask.
> 
> BTW, it's late here, and I haven't had time to try any meshes with it.
> Please let me know if you run into problems doing so (smooth triangles
> should convert without problems, but you never know).

I have some code (untested--Rusty takes priority!) that implements
a quartic triangular Bezier patch.  There is supposedly an algorithm
by which a Loop subdivision surface can be converted into a bunch of
these (which makes them much easier to implement).

Regards,
John


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From: Ben Chambers
Subject: Re: Curved Triangles
Date: 27 Jun 2002 20:56:51
Message: <3d1bb453@news.povray.org>
"John VanSickle" <evi### [at] hotmailcom> wrote in message
news:3D1B59E3.54F30927@hotmail.com...
> > BTW, it's late here, and I haven't had time to try any meshes with it.
> > Please let me know if you run into problems doing so (smooth triangles
> > should convert without problems, but you never know).
>
> I have some code (untested--Rusty takes priority!) that implements
> a quartic triangular Bezier patch.  There is supposedly an algorithm
> by which a Loop subdivision surface can be converted into a bunch of
> these (which makes them much easier to implement).

All the effort I put into it, and the code already exists somewhere
else?!?!?  Aaaaargh!

That's what I get for not researching the topic first!

BTW, where is that code you were talking about?

...Chambers


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From: John VanSickle
Subject: Re: Curved Triangles
Date: 29 Jun 2002 10:57:20
Message: <3D1DCAD0.90317117@hotmail.com>
Ben Chambers wrote:
> 
> "John VanSickle" <evi### [at] hotmailcom> wrote in message
> news:3D1B59E3.54F30927@hotmail.com...
> > > BTW, it's late here, and I haven't had time to try any meshes
> > > with it. Please let me know if you run into problems doing so
> > > (smooth triangles should convert without problems, but you never
> > > know).
> >
> > I have some code (untested--Rusty takes priority!) that implements
> > a quartic triangular Bezier patch.  There is supposedly an algorithm
> > by which a Loop subdivision surface can be converted into a bunch of
> > these (which makes them much easier to implement).
> 
> All the effort I put into it, and the code already exists somewhere
> else?!?!?  Aaaaargh!
> 
> That's what I get for not researching the topic first!

Yup.  Google is wonderful, but only if you use it.

> BTW, where is that code you were talking about?

Here is the UNTESTED code:

// start of macro code
#macro TriBezier(pA,pB,pC,pD,pE,pF,pG,pH,pI,pJ,pK,pL,pM,pN,pO,cD)
  #if(cD<1)
    smooth_triangle {
      A,vcross(C-A,B-A),
      K,vcross(G-K,L-K),
      O,vcross(N-O,J-O)
    }
  #else
TriBezier(A,(A+B)/2,(A+C)/2,(A+2*B+D)/4,(A+B+C+E)/4,(A+2*C+F)/4,
  (A+3*B+3*D+G)/8,(A+2*B+D+C+2*E+H)/8,(A+2*C+F+B+2*E+I)/8,
  (A+3*C+3*F+J)/8,(A+4*B+6*D+4*G+K)/16,(A+3*B+3*D+G+C+3*E+3*H+L)/16,
  (A+2*C+F+2*B+4*E+2*I+D+2*H+M)/16,(A+3*C+3*F+J+B+3*E+3*I+N)/16,
  (A+4*C+6*F+4*J+O)/16,cD-1)

TriBezier((A+4*B+6*D+4*G+K)/16,(B+3*D+3*G+K)/8,(B+C+3*D+3*E+3*G+3*H+K+L)/16,
  (D+2*G+K)/4,(D+E+2*G+2*H+K+L)/8,(D+2*E+F+2*G+4*H+2*I+K+2*L+M)/16,
  (G+K)/2,(G+H+K+L)/4,(G+2*H+I+K+2*L+M)/8,(G+3*H+3*I+J+K+3*L+3*M+N)/16,
  K,(K+L)/2,(K+2*L+M)/4,(K+3*L+3*M+N)/8,(K+4*L+6*M+4*N+O)/16,cD-1)

TriBezier((A+4*C+6*F+4*J+O)/16,(B+C+3*E+3*F+3*I+3*J+N+O)/16,
  (C+3*F+3*J+O)/8,(D+2*E+F+2*H+4*I+2*J+M+2*N+O)/16,(E+F+2*I+2*J+N+O)/8,
  (F+2*J+O)/4,(G+3*H+3*I+J+L+3*M+3*N+O)/16,(H+2*I+J+M+2*N+O)/8,
  (I+J+N+O)/4,(J+O)/2(K+4*L+6*M+4*N+O)/16,(L+3*M+3*N+O)/8,(M+2*N+O)/4,
  (N+O)/2,O,cD-1)

TriBezier((A+4*B+6*D+4*G+K)/16,(A+3*B+3*D+G+C+3*E+3*H+L)/16,
  (A+2*B+D+2*C+4*E+2*H+F+2*I+M)/16(A+3*C+3*F+J+B+3*E+3*I+N)/16,
  (A+4*C+6*F+4*J+O)/16,(B+C+3*D+3*E+3*G+3*H+K+L)/16,
  (B+C+2*D+3*E+F+G+3*H+2*I+L+M)/16,(B+C+D+3*E+2*F+2*H+3*I+J+M+N)/16,
  (B+C+3*E+3*F+3*I+3*J+N+O)/16,(D+2*E+F+2*G+4*H+2*I+K+2*L+M)/16,
  (D+2*E+F+G+3*H+3*I+J+L+2*M+N)/16,(D+2*E+F+2*H+4*I+2*J+M+2*N+O)/16,
  (G+3*H+3*I+J+K+3*L+3*M+N)/16,(G+3*H+3*I+J+L+3*M+3*N+O)/16,
  (K+4*L+6*M+4*N+O)/16,cD-1)
  #end
#end

// end of macro code

I am also generalizing my surface subdivision code to include four-
sided faces (which helps solve a couple of problems with uv-mapping).
That project isn't done yet.

Regards,
John


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From: Ben Chambers
Subject: Re: Curved Triangles
Date: 29 Jun 2002 16:49:45
Message: <3d1e1d69@news.povray.org>
"John VanSickle" <evi### [at] hotmailcom> wrote in message
news:3D1DCAD0.90317117@hotmail.com...
> Here is the UNTESTED code:
>
> // start of macro code
> #macro TriBezier(pA,pB,pC,pD,pE,pF,pG,pH,pI,pJ,pK,pL,pM,pN,pO,cD)
>   #if(cD<1)
>     smooth_triangle {
>       A,vcross(C-A,B-A),
>       K,vcross(G-K,L-K),
>       O,vcross(N-O,J-O)
>     }
>   #else
> TriBezier(A,(A+B)/2,(A+C)/2,(A+2*B+D)/4,(A+B+C+E)/4,(A+2*C+F)/4,
>   (A+3*B+3*D+G)/8,(A+2*B+D+C+2*E+H)/8,(A+2*C+F+B+2*E+I)/8,
>   (A+3*C+3*F+J)/8,(A+4*B+6*D+4*G+K)/16,(A+3*B+3*D+G+C+3*E+3*H+L)/16,
>   (A+2*C+F+2*B+4*E+2*I+D+2*H+M)/16,(A+3*C+3*F+J+B+3*E+3*I+N)/16,
>   (A+4*C+6*F+4*J+O)/16,cD-1)
>
>
TriBezier((A+4*B+6*D+4*G+K)/16,(B+3*D+3*G+K)/8,(B+C+3*D+3*E+3*G+3*H+K+L)/16,
>   (D+2*G+K)/4,(D+E+2*G+2*H+K+L)/8,(D+2*E+F+2*G+4*H+2*I+K+2*L+M)/16,
>   (G+K)/2,(G+H+K+L)/4,(G+2*H+I+K+2*L+M)/8,(G+3*H+3*I+J+K+3*L+3*M+N)/16,
>   K,(K+L)/2,(K+2*L+M)/4,(K+3*L+3*M+N)/8,(K+4*L+6*M+4*N+O)/16,cD-1)
>
> TriBezier((A+4*C+6*F+4*J+O)/16,(B+C+3*E+3*F+3*I+3*J+N+O)/16,
>   (C+3*F+3*J+O)/8,(D+2*E+F+2*H+4*I+2*J+M+2*N+O)/16,(E+F+2*I+2*J+N+O)/8,
>   (F+2*J+O)/4,(G+3*H+3*I+J+L+3*M+3*N+O)/16,(H+2*I+J+M+2*N+O)/8,
>   (I+J+N+O)/4,(J+O)/2(K+4*L+6*M+4*N+O)/16,(L+3*M+3*N+O)/8,(M+2*N+O)/4,
>   (N+O)/2,O,cD-1)
>
> TriBezier((A+4*B+6*D+4*G+K)/16,(A+3*B+3*D+G+C+3*E+3*H+L)/16,
>   (A+2*B+D+2*C+4*E+2*H+F+2*I+M)/16(A+3*C+3*F+J+B+3*E+3*I+N)/16,
>   (A+4*C+6*F+4*J+O)/16,(B+C+3*D+3*E+3*G+3*H+K+L)/16,
>   (B+C+2*D+3*E+F+G+3*H+2*I+L+M)/16,(B+C+D+3*E+2*F+2*H+3*I+J+M+N)/16,
>   (B+C+3*E+3*F+3*I+3*J+N+O)/16,(D+2*E+F+2*G+4*H+2*I+K+2*L+M)/16,
>   (D+2*E+F+G+3*H+3*I+J+L+2*M+N)/16,(D+2*E+F+2*H+4*I+2*J+M+2*N+O)/16,
>   (G+3*H+3*I+J+K+3*L+3*M+N)/16,(G+3*H+3*I+J+L+3*M+3*N+O)/16,
>   (K+4*L+6*M+4*N+O)/16,cD-1)
>   #end
> #end
>
> // end of macro code

Hmm, seems just a little bit messy.  Where on the triangle are the control
points placed?  (One thing I wanted to achieve with mine is that it uses no
more data than the original triangle - three points, and three normals, so
control points don't have to be generated).

...Chambers


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From: John VanSickle
Subject: Re: Curved Triangles
Date: 30 Jun 2002 08:02:12
Message: <3D1EF346.F0631240@hotmail.com>
Ben Chambers wrote:
> 
> Hmm, seems just a little bit messy.  Where on the triangle are the
> control points placed?  (One thing I wanted to achieve with mine is
> that it uses no more data than the original triangle - three points,
> and three normals, so control points don't have to be generated).

There is a site that has a PDF describing how to get the control
points for a cubic Bezier triangular patch, derived from the corners
and normals of that corner.  It's here:

  http://www-courses.cs.uiuc.edu/~cs319/subdiv.pdf

It won't handle sharp edges (which are important for most work). Note
that this is for CUBIC patches, whereas the code I gave above is for
quartic patches.  For cubic patches, you'd need this macro:

// start of macro code
#macro CubicTriPatch(A,B,C,D,E,F,G,H,I,J,cD)

#if(cD<1)
  smooth_triangle { A,vcross(C-A,B-A),
    G,vcross(D-G,H-G),J,vcross(I-J,F-J) }
#else
  CubicTriPatch(A,(A+B)/2,(A+C)/2,(A+2*B+D)/4,(A+B+C+E)/4,(A+2*C+F)/4,
    (A+3*B+3*D+G)/8,(A+2*B+D+C+2*E+H)/8,(A+B+2*C+2*E+F+I)/8,
    (A+3*C+3*F+J)/8,cD-1 )

  CubicTriPatch(G,(G+H)/2,(D+G)/2,(G+2*H+I)/4,(D+E+G+H)/4,(B+2*D+G)/4,
    (G+3*H+3*I+J)/8,(D+2*E+F+G+2*H+I)/8,(G+H+2*D+2*E+B+C)/8,
    (A+3*B+3*D+G)/8,cD-1 )

  CubicTriPatch(J,(J+F)/2,(I+J)/2,(J+2*F+C)/4,(E+F+I+J)/4,(H+2*I+J)/4,
    (A+3*C+3*F+J)/8,(I+2*E+B+J+2*F+C)/8,(J+F+2*I+2*E+H+D)/8,
    (G+3*H+3*I+J)/8,cD-1 )

  CubicTriPatch((A+3*B+3*D+G)/8,(A+2*B+D+C+2*E+H)/8,
    (A+B+2*C+2*E+F+I)/8,(A+3*C+3*F+J)/8,(G+H+2*D+2*E+B+C)/8,
    (B+C+D+2*E+F+H+I)/8,(I+2*E+B+J+2*F+C)/8,(D+2*E+F+G+2*H+I)/8,
    (D+2*E+F+H+2*I+J)/8,(G+3*H+3*I+J)/8,cD-1 )
#end
#end

// end of macro code

Regards,
John


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