Re: [eigen] Rigid transformations in eigen: use of dual quaternions |

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*To*: eigen@xxxxxxxxxxxxxxxxxxx*Subject*: Re: [eigen] Rigid transformations in eigen: use of dual quaternions*From*: Rohit Garg <rpg.314@xxxxxxxxx>*Date*: Sat, 12 Sep 2009 23:36:36 +0530*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:in-reply-to:references :date:message-id:subject:from:to:content-type; bh=yRgL0C5ByhVow3qlH+LZUhNWZa9iBnkfO295obtZOgQ=; b=lHdIuTgZaDHlgf0GGfdXkHFVQ9PO8i8LN0+OUBYCJVeLnfxn7lzT4WcRM0zOLh01Uj SvPyN2Nun5R2WwW2dpHNGAIuZwDl34S8q7ng+QYuBnuX6NMW5AhrlTn2B6Dt5aAtr0sP +XbNQ8wFtRfr0I5j7AzKlZLfbdaEQPPVrZzd0=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:in-reply-to:references:date:message-id:subject:from:to :content-type; b=qezGfWgHEsWFpy5o8p7uuego1wlv1hDtzKUU18A6+FXeSE5asQzvu6Zgb0PMUocA2t kMNDrVMT9ySF6M53tPV/1IIYK4BbK43xnV1IN2DJtvpuW034qMbau7EQgH9DemzziQyh VBbdeQ0nGvwCpgqsn7aPGsRMbgzA9DnHU6PrA=

>> For what-are-dual-quaternions, look at the paper here >> >> http://isg.cs.tcd.ie/projects/DualQuaternions/ > > Hm that page wasn't very explicit, but I found this: > > http://en.wikipedia.org/wiki/Dual_quaternion Must have been taken down. Paper attached. > > Now I understand that this notion makes a lot of sense and is useful > to have in Geometry. They are also simple to describe theoretically, > as they're of the form q1 + epsilon * q2 where q1,q2 are quaternions > and epsilon is subject to the algebraic rule epsilon^2=0. > > Quote from the wikipedia: "Similar to the way that rotations in 3D > space can be represented by quaternions of unit length, rigid motions > in 3D space can be represented by dual quaternions of unit length." > > So i'd say, green light to add a dual quaternion class to Geometry. > >> Rigid transformation is transformation that preserves a rigid body (ie >> distance and angle preserving). Translation, rotations, reflections >> do. Scaling and shearing don't. > > Oh, I see. So what they call a "rigid transformation" is what's been > called an isometry for a century. Why do computer scientists have to > rename everything ? :) Physicists call it rigid transformation too... :) >> All in all, dual quaternions are to rigid transformations what >> quaternions are to 3D rotations. The biggest advantage is to treat >> rotation and translation in a unified framework. > > If it were just that, we have the Transform class. But I understand > that the dual quaternion representation allows for that interesting > slerp-like interpolation, i can believe it's useful, and dual > quaternions have a wikipedia page mentioning applications to 3D > graphics, so, no need to convince me any more than that. I'd say, go > for it! The advantages are cheaper to store, cheaper to compose, more stable and interpolation. In that sense, they correspond to the advantages of quaternions over matrices. Question: can you map a piece of memory as a Quaternion<datatype> (with vectorization)? Like the question I raised earlier today where a vec4i wasn't vectorized. If you can, it is trivial to do with all the expression template goodness. Even I'll be able to do it then. :) -- Rohit Garg http://rpg-314.blogspot.com/ Senior Undergraduate Department of Physics Indian Institute of Technology Bombay

**Attachment:
dualQuats.pdf**

**Follow-Ups**:**Re: [eigen] Rigid transformations in eigen: use of dual quaternions***From:*Benoit Jacob

**References**:**[eigen] Rigid transformations in eigen: use of dual quaternions***From:*Rohit Garg

**Re: [eigen] Rigid transformations in eigen: use of dual quaternions***From:*Benoit Jacob

**Re: [eigen] Rigid transformations in eigen: use of dual quaternions***From:*Rohit Garg

**Re: [eigen] Rigid transformations in eigen: use of dual quaternions***From:*Benoit Jacob

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