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This book concerns the generation and improvement of freeform motions. Aside from an extensive review of the existing work, novel techniques for motion construction and improvement are developed, with numerical and pictorial evidence attesting to their suitability. In the latter part of this thesis, a non-standard geometric algebra called G-4 is described and used for motion representation, and it is shown to be capable and economical. Whilst this document is theoretical in nature, it may be useful in computer animation and robotics.