NDLI: Linear combination of transformations

Content Provider	ACM Digital Library
Author	Alexa, Marc
Copyright Year	2002
Abstract	Geometric transformations are most commonly represented as square matrices in computer graphics. Following simple geometric arguments we derive a natural and geometrically meaningful definition of scalar multiples and a commutative addition of transformations based on the matrix representation, given that the matrices have no negative real eigenvalues. Together, these operations allow the linear combination of transformations. This provides the ability to create weighted combination of transformations, interpolate between transformations, and to construct or use arbitrary transformations in a structure similar to a basis of a vector space. These basic techniques are useful for synthesis and analysis of motions or animations. Animations through a set of key transformations are generated using standard techniques such as subdivision curves. For analysis and progressive compression a PCA can be applied to sequences of transformations. We describe an implementation of the techniques that enables an easy-to-use and transparent way of dealing with geometric transformations in graphics software. We compare and relate our approach to other techniques such as matrix decomposition and quaternion interpolation.
Starting Page	380
Ending Page	387
Page Count	8
File Format	PDF
ISSN	07300301
e-ISSN	15577368
DOI	10.1145/566654.566592
Volume Number	21
Issue Number	3
Journal	ACM Transactions on Graphics (TOG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2002-07-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Exponential map Linear space Logarithm Matrix exponential Transformations
Content Type	Text
Resource Type	Article
Subject	Computer Graphics and Computer-Aided Design

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