On 8 Dec 1999 09:22:48 -0500, Nieminen Juha wrote:
>Ron Parker <ron### [at] povrayorg> wrote:
>: There is a paper out there somewhere - can't remember whether I saw it on the 
>: web or in an old ACM publication - that talks about methods for tracing curved
>: rays.  It's still not perfect, because it essentially traces the curves as a
>: series of line segments, but at least it doesn't require tesselation.
>
>  Perhaps this kind of non-uniform transformation could be possible with
>povray?
>
>  Btw, why is not possible to scale an object so that the scale changes
>linearly along an axis? I don't remember the name if this transformation,
>but it's the one that can convert a cylinder into a cone.
>  I think that it's a linear transformation. I line transformed this way
>is still a line after the transformation. Should be perfectly possible
>in povray. If this is so, wouldn't it be a good idea to add this kind
>of transformation?

It's a perspective transformation.  It's not linear, in that 
f(A)+f(B) != f(A+B).  I'm not sure it transforms lines to lines, 
either.  Consider the (2d) perspective transformation f(x,y) = xy.  
If you transform a line parallel to the X axis, such as y=2, then 
you get a line in return (y=2x).  If you transform a line parallel 
to the Y axis, you also get a line in return.  But what happens 
when you transform the line y=x?  By my calculations, you get y=x*x, 
which is not a line.

For another thought experiment, assume that you have a transformation 
that can turn a cylinder into a cone.  What would it do to the 
resulting cone?  If it leaves it a cone, then the transformation 
isn't invertible, because there's some cylinder that maps to the same 
cone.  If it makes it anything else, then what does it do to a line 
that lies in the surface of that cone?

-- 
These are my opinions.  I do NOT speak for the POV-Team.
The superpatch: http://www2.fwi.com/~parkerr/superpatch/
My other stuff: http://www2.fwi.com/~parkerr/traces.html

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