Abstract
Every nonsingular linear transformation of three-dimensional space is the product of three scales, two shears, and one rotation. This chapter explains how to decompose any arbitrary, singular or nonsingular, linear, or affine transformation of three-dimensional space into simple, geometrically meaningful factors. Linear transformations of three-dimensional space are generally represented by 3 × 3 matrices. The parallel projection can be replaced by a shear followed by an orthogonal projection. Every nonsingular affine transformation of three-dimensional space can be factored into the product of three scales, two shears, one rotation, and one translation. Similarly, every singular affine transformation of three-dimensional space can be factored into the product of two scales, two shears, one rotation, one orthogonal projection, and one translation. Those decompositions are not unique because the decompositions of the associated linear transformations are not unique. Those procedures are still of some value as they allow to decompose arbitrary affine transformations into simple, geometrically meaningful factors.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
