[Insight-users] Translation, Rotation and Scaling

[Luis Ibanez]

Hi Ramon,

I'm not quite persuaded by your demonstration.  :-)

First of all, your image: "orig-scaling-rotation.jpg"

doesn't seem to have any rotation applied to it...

Are you sure that you attached the correct image ?

Second:
Yes,
I agree with you, in that the order of operations,
is important.


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...


    Regards,


          Luis


-------------------------
2009/9/9 Ramón Casero Cañas <ramon.casero at comlab.ox.ac.uk>:
> Luis Ibanez wrote:
>>
>> Hi Ramon,
>>
>> When you have Anisotropic Scaling and Rotation,
>> that is equivalent to providing Shearing.
>>
>> In other words, you can implement Shearing by
>> composing Rotation and Anisotropic scaling.
>>
>> Therefore, you cannot have a transform that has
>> Anisotropic Scaling and Rotation but not Shearing.
>
>
> Hi Luis,
>
> Thanks for your reply. I hope I'm not digging a hole for myself :), but I
> think in this case the order of composition is important. I have attached 3
> pictures to illustrate this.
>
> orig.jpg is the original image.
>
> I agree that if you have rotation followed by anisotropic scaling, in
> general you get shearing. For example, rotating the figure -45º, scaling the
> width by 1/2 and then rotating back (NB. not by 45º) produces the sheared
> figure orig-rotation-scaling.jpg.
>
> However, if you have anisotropic scaling followed by rotation, then you
> cannot get shearing, as illustrated by figure orig-scaling-rotation.jpg.
>
> Thus, there has to be a transformation with both anisotropic scaling and
> rotation but not shearing, hasn't it?
>
> Best regards,
>
> Ramon.
>
> --
> Ramón Casero Cañas, DPhil
>
> Computational Biology, Computing Laboratory
> University of Oxford
> Wolfson Building, Parks Rd
> Oxford OX1 3QD
>
> tlf     +44 (0) 1865 610807
> web     http://web.comlab.ox.ac.uk/people/Ramon.CaseroCanas
> photos  http://www.flickr.com/photos/rcasero/
>