How to apply say 1/2 or 1/3 of a given 3D matrix effect ?

[ggaliens]

How to apply say 1/2 or 1/3 of a given 3D matrix effect ?

I have a composite matrix that involves both rotation and translation ... (in point of fact) ... but it would also be cool if we would consider it to contain scaling aspects as well ...

I use this matrix to move and change orientation of an object from one place to another (don't forget it is rotated). The API I use ... also creates a nice inverse matrix during the composition phase ... but this may be an aside.

Without getting into deconstructing the various components of the transformation back to rotation and translation ... how could I very simply apply a percentage of the given transformation ... call it "M1" ?

 

[Ishuda]

How to apply say 1/2 or 1/3 of a given 3D matrix effect ?

I have a composite matrix that involves both rotation and translation ... (in point of fact) ... but it would also be cool if we would consider it to contain scaling aspects as well ...

I use this matrix to move and change orientation of an object from one place to another (don't forget it is rotated). The API I use ... also creates a nice inverse matrix during the composition phase ... but this may be an aside.

Without getting into deconstructing the various components of the transformation back to rotation and translation ... how could I very simply apply a percentage of the given transformation ... call it "M1" ?

In order to make it easier to give an example, I'll only work in 2D but the same sort of thing applies in 3D (or higher dimensions) only more complicated because there are more choice of axis to rotate about.

The general rotation, scaling,shearing, and translation matrix for a 2D system can be written as
\(\displaystyle (u,\, v) = (x\, -x_0, y\, -y_0) \begin{pmatrix} a\, cos(\theta)& - b\, sin(\theta)\\ c\, sin(\theta)& d\, cos(\theta) \end{pmatrix}\)
where (x₀, y₀) is the translation [the new origin], a, b, c, and d represent the scaling and shear, and \(\displaystyle \theta\) representing the rotation.

As you can see there is not a simple relationship between any general matrix T
\(\displaystyle T = \begin{pmatrix} A& B\\ C& D\end{pmatrix}\)
and the above transformation matrix. So the first thing one must ask is 1/2 (1/3, ...) of what? The scaling factor? The rotation angle? The x shear factor? The y shear factor? The translation?

The general composition of matrices and transformations can be found around the web. If you wanted to start with 2D, you could certainly do worse than
http://www.willamette.edu/~gorr/classes/GeneralGraphics/Transforms/transforms2d.htm