Re: [Algorithms] Texel area

[Diogo de A. <dio...@sp...>]

Hi Olivier,

I was already having similar thoughts on how to solve this (use the
barycentric coordinates to solve a linear system), so it seems to me that
you got the jist of it, I'm going to try implementing... 

I'm just unclear on a couple of things:

> That's a rather usual linear system.  Noting the inverse determinant

How does this matter? Not questioning it, just trying to understand what
follows from here...

> You'll notice that all these are linear systems, which means a texel
rectangle size is independant of its position.  

This seems counter-intuitive... I understand your reasoning, and I can't
find a mistake in your logic, but it seems to me that the size shouldn't be
constant (from an intuitive standpoint)...

Best regards,
Diogo


-----Original Message-----
From: Olivier Galibert [mailto:gal...@po...] 
Sent: sexta-feira, 4 de Fevereiro de 2011 19:32
To: gda...@li...
Subject: Re: [Algorithms] Texel area

On Fri, Feb 04, 2011 at 10:50:08AM -0000, Diogo de Andrade wrote:
> So, what I want to do is, given
> 
> - triangle T=(V1,V2,V3), in which V1, V2, V3 are vertexes that have 
> some properties (world space position, normal, diffuse color, texture 
> coordinates
> 0 (in uniform space), texture coordinates 1 (in texture space, 
> [0..texture size[), etc),
> 
> - texture size
> 
> - position inside the triangle (absolute value in texture space)
> 
> is to find out the rectangle that bounds the texel in world space 
> (origin+2 vectors)...

If that's your real problem, the solution is reasonably simple and has
nothing to do with the rasterization.

You have three points when space/texture coordinates:
  (x0, y0, z0, u0, v0)
  (x1, y1, z1, u1, v1)
  (x2, y2, z2, u2, v2)

Given u, v the position in texture space, you first want to find the
barycentric coordinates (t1, t2):
 | u0 + t1*(u1-u0) + t2*(u2-u0) = u
 | v0 + t1*(v1-v0) + t2*(v2-v0) = v

That's a rather usual linear system.  Noting the inverse determinant
di:
   di = 1/((v2-v0)*(u1-u0) - (u2-u0)*(v1-v0))

then:
 | t1 = di*((v2-v0)*(u-u0) - (u2-u0)*(v-v0))  | t2 = di*((v1-v0)*(u-u0) -
(u1-u0)*(v-v0))

>From these, you can easily find the world coordinates:
 | x = x0 + t1*(x1-x0) + t2*(x2-x0)
 | y = y0 + t1*(y1-y0) + t2*(y2-y0)
 | z = z0 + t1*(z1-z0) + t2*(z2-z0)

You'll notice that all these are linear systems, which means a texel
rectangle size is independant of its position.  So you can get your vector
once by computing the position differences between (0, 0), (0.5, 0) and (0,
0.5) for instance.  You'll notice the equations simplify nicely when
computing deltas if you do the math.

I suspect it's not your real problem though, so feel free to correct it :-)

  OG.


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