Here is the "projective transform" I'm finally writing about...
It includes 8 parameters (Rx,Ry,Rz,Tx,Ty,Tz,Sx,Sy) which are
present in POV-Ray. I use it to represent the motion of cameras.
Francois LE COAT writes:
> When the raytracing POV-Ray program projects a 3D movement in a 2D
> image sequence, it uses what is called "projective transformation" that
> produces the images. This projective transform is well known in the
> literature, and is called "homography". It takes the aspect of :
m01, m02) (u)
11, m12) (v)
0, m21, m22) (1)
> where (u,v) is the coordinates in the first image, and (up,vp) its
> movement's transform correspondent in the second image, in homogeneous
> notations. This projective transformation takes 9 parameters (mij),
> but you can consider that it takes only 8 parameters, if those are
> normalized (one of the parameters (mij) is forced to 1.0 value).
> If you consider those 8 unique parameters, because it allows to
> project a 3D movement, in 2D images, are linked to the 6 degrees
> of freedom in 3D space, via the 3 Euler angles in rotations, and
> 3 translations. Those (mij) parameters are linked to :
> (Tx, Ty, Tz) the 3 translations in pixels of images, and
> (Rx, Ry, Rz) the (yaw, pitch, roll) Euler angles in degrees.
> When someone like me, uses POV-Ray scripting capabilities, I have
> the following directives in the language that represent the movement :
> translate <Tx, Ty, Tz> and
> rotate <Rx, Ry, Rz>
> in the 3D space. But these directives only takes account of 6 parameter
> of the projective transform, and I miss 2 more parameters in "skewness"
> This is well represented in the above document. When I consider a
> movement in 3D space, projected in a 2D sequence of images, I must
> consider two other parameters (Sx,Sy) in (x,y) coordinates of image,
> that are linked to (x,y) skewness. My question is, how are those skew
> angles present in the POV-Ray scripting language ? There should be a
> skew <Sx, Sy>
> directive that would render the skewness type of movement of the camera
> I can't figure out myself, how these 2 skew movements are represented
> in the POV-Ray scripting language directives. It must be linked to
> the way how the perspective is generated in the camera properties.
> Can somebody tell me how to generate those parameter movements with
> POV-Ray language ? I'm confused, and probably not very clear, but maybe
> someone else has thought about this problem, before me.
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