conformal decomposition

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alexander arsenovic

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May 23, 2017, 11:49:08 AM5/23/17
to Geometric_Algebra
is there a method to decompose a  conformal transformation into a set of elementary transformations, using CGA? from hestenes's "design of linear algebra and geometry" he gives several decomposition, eq 5.29, 5.33 for the group in G(1,1) and another in eq 5.89 for G(n+1,1), but it is not clear to me how to implement the decomposition. 

for example, if i have the CGA verser for a translation+rotation, can  i separate the rotation from the translation? more generally, i want to decompose  an arbitrary conformal transformation into a set of elementary transformations. 

here is a notebook demonstrating the question in context of an application,
 
thanks!
alex

Pablo Colapinto

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May 23, 2017, 1:18:30 PM5/23/17
to Geometric_Algebra
Hi Alex,

I think you would enjoy taking a look at Dorst and Valkenburg "Square Roots and Logarithm of Rotors" paper: http://link.springer.com/chapter/10.1007/978-0-85729-811-9_5 which details how to use the Bivector split introduced by Hestenes and Sobcyzk in "Clifford Algebra to Geometric Calculus" to decompose general conformal transformations into simple bivector exponentials.  This can definitely be used to decompose a motor into rotational and translational components, but it can do even more. Dorst's "Construction of 3D Conformal Motions" is a more recent version: http://link.springer.com/article/10.1007/s11786-016-0250-8.

Also, in case it is useful, I have a paper that uses his decomposition technique to compose cyclidic nets as a tensor product of two simple bivector exponentials: https://www.academia.edu/33088139/Composing_Surfaces_with_Conformal_Rotors 
and a chapter in my thesis explores decomposing circle to circle transformations (which are the most general in 5D cga).

Hope this is helpful!
Pablo

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alexander arsenovic

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May 25, 2017, 10:21:22 AM5/25/17
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thanks  a lot pablo,  i like your thesis. 

after looking some more  into  those papers,  its not clear that they answer my question. my understanding is that they show how to break up a Rotor into commuting simple rotors. but  my question  is how to  decompose an arbitrary verser into a given  conical form. since each operation is a rotations in CGA, this might be the same question as decomposing a rotation into a set of rotations in given planes ( i asked this questions a while ago, here). generally speaking , the decomposition problem seems to me to be related to Lie group decomposition, 

in either case, i have come up with a solution to my original question , here, but it seems a little awkward to me. id like  to find a more straightforward approach, that doesnt involve 'probing' a transform with known vectors, but maybe this is the way to do it, i dont know. 

On Tue, May 23, 2017 at 1:18 PM, Pablo Colapinto <wolf...@gmail.com> wrote:
Hi Alex,

I think you would enjoy taking a look at Dorst and Valkenburg "Square Roots and Logarithm of Rotors" paper: http://link.springer.com/chapter/10.1007/978-0-85729-811-9_5 which details how to use the Bivector split introduced by Hestenes and Sobcyzk in "Clifford Algebra to Geometric Calculus" to decompose general conformal transformations into simple bivector exponentials.  This can definitely be used to decompose a motor into rotational and translational components, but it can do even more. Dorst's "Construction of 3D Conformal Motions" is a more recent version: http://link.springer.com/article/10.1007/s11786-016-0250-8.

Also, in case it is useful, I have a paper that uses his decomposition technique to compose cyclidic nets as a tensor product of two simple bivector exponentials: https://www.academia.edu/33088139/Composing_Surfaces_with_Conformal_Rotors 
and a chapter in my thesis explores decomposing circle to circle transformations (which are the most general in 5D cga).

Hope this is helpful!
Pablo

On Tue, May 23, 2017 at 8:49 AM alexander arsenovic <al...@810lab.com> wrote:
is there a method to decompose a  conformal transformation into a set of elementary transformations, using CGA? from hestenes's "design of linear algebra and geometry" he gives several decomposition, eq 5.29, 5.33 for the group in G(1,1) and another in eq 5.89 for G(n+1,1), but it is not clear to me how to implement the decomposition. 

for example, if i have the CGA verser for a translation+rotation, can  i separate the rotation from the translation? more generally, i want to decompose  an arbitrary conformal transformation into a set of elementary transformations. 

here is a notebook demonstrating the question in context of an application,
 
thanks!
alex

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