Yesterday (lecture 1) – I briefly talked about Linear Combinations of Transformations. If you want to see more about them, or get them in a less hand-wavy way, here are some references.
The “original” paper that introduced the idea to Computer Graphics:
- Marc Alexa. Linear combination of transformations (2002) ACM Transactions on Graphics 21 (3) p. 380-380-387-387 http://dl.acm.org/citation.cfm?id=566654.566592
Here is a rant by a couple of well known, and very smart, game developers who know a lot of math and practical stuff. I think the truth is somewhere in between. And, you wouldn’t want to use matrix exponentials just for rotations (since there are special case techniques for them).
- Charles Bloom, Jonathan Blow, and Casey Muratori. Errors and Omissions in Marc Alexa’s “Linear Combination of Transformations”. (web page)
Cindy Grimm and her students worked out some details for computing the matrix logs and exponentials efficiently, especially for the 4×4 case.
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Keyframing using Linear Interpolation of Matrices
. Amy Hawkins and Cindy Grimm. Journal of Graphics Tools, 12(13): 55-69, 2007. [PDF] [BibTeX] [Web]
Finally, here are 3 things I did using Matrix Exponentials (to prove they actually have some application):
- Effective Replays and Summarization of Virtual Experiences
Kevin Ponto, Joe Kohlmann, Michael Gleicher. IEEE Transactions on Visualization and Computer Graphics, Volume 18, Number 4 – April 2012 (web) - Automated Illustration of Molecular Flexibility . Aaron Bryden, George Phillips Jr., Michael Gleicher. IEEE Transactions on Visualization and Computer Graphics, Volume 18, Number 1, page 132–145 – Jan 2012 (web).
- Re-Cinematography: Improving the Camerawork of Casual Video . Michael Gleicher, Feng Liu. ACM Transactions on Multimedia Computing Communications and Applications (TOMCCAP), Volume 5, Number 1, page 1–28 – 2008 (web)
