[Insight-users] Translation, Rotation and Scaling

[Ramón Casero Cañas]

Luis Ibanez wrote:
> Hi Ramon,
> 
> I'm not quite persuaded by your demonstration.  :-)

Hi Luis,

That's fine, because what you are saying is right :)


> First of all, your image: "orig-scaling-rotation.jpg"
> 
> doesn't seem to have any rotation applied to it...

Yes, it's not very clear. In orig-rotation-scaling.jpg, there's a
rotation, scaling, and another rotation to go back to a landscape image.

In orig-scaling-rotation.jpg, there's a scaling, and then the rotation
and counter rotation cancel each other.

What I wanted to illustrate, and it seems that I didn't make it very
clear, is that with a scaling followed by rotation, one cannot make the
buildings skewed to one side (in the sense of losing the right angles).


> Third:
> A Transform has Shearing if it doesn't preserve
> the angles between straight lines.
> 
> A Transform that applies anisotropic scaling,
> indeed, doesn't preserve angles.
> 
> You may be getting too focused on treating
> the X,Y axis as if they were special lines in
> the plane.  You probably will see this clearer
> if you think about the geometric Transformation
> in terms of their effects on basic geometric
> figures such a square, a circle, two lines at
> an angle and so on...


The not slanted buildings is what I had in mind when talking about
shear.

However, you are right. Although scaling + rotation will preserve right
angles parallel to the X, Y axis (and thus the squareness of the picture 
frame), it changes other angles.

Thanks for the discussion, it was very interesting.


A derived question would then be, according to the discussion, with 5 
parameters (2 scalings + 2 translations + 1 rotation) it seems that you 
can get all in an affine transformations. But the affine transformation 
has 6 parameters. Is it redundant in one parameter? Is there something 
I'm missing?


Ramon.

-- 
Ramón Casero Cañas, DPhil

Computational Biology, Computing Laboratory
University of Oxford
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