Geometric algebra as applied to freeform motion design and improvement

Freeform curve design has existed in various forms for at least two millennia, and is important throughout computer-aided design and manufacture. With the increasing importance of animation and robotics, coupled with the increasing power of computers, there is now interest in freeform motion design, which, in part, extends techniques from curve design, as well as introducing some entirely distinct challenges. There are several approaches to freeform motion construction, and the first step in designing freeform motions is to choose a representation. Unlike for curves, there is no "standard" way of representing freeform motions, and the different tools available each have different properties. A motion can be viewed as a continuously-varying pose, where a pose is a position and an orientation. This immediately presents a problem; the dimensions of rotations and translations are different, and it is not clear how the two can be compared, such as to define distance along a motion. One solution is to treat the rotational and translational components of a motion separately, but this is inelegant and clumsy. The philosophy of this thesis is that a motion is not defined purely by rotations and translations, but that the body following a motion is a part of that motion. Specifically, the part of the body that is accounted for is its inertia tensor. The significance of the inertia tensor is that it allows the rotational and translational parts of a motion to be, in some sense, compared in a dimensionally- consistent way. Using the inertia tensor, this thesis finds the form of kinetic energy in <;1'4, and also discusses extensions of the concepts of arc length and curvature to the space of motions, allowing techniques from curve fairing to be applied to motion fairing. Two measures of motion fairness are constructed, and motion fairing is the process of minimizing the measure of a motion by adjusting degrees of freedom present in the motion's construction. This thesis uses the geometric algebra <;1'4 in the generation offreeform motions, and the fairing of such motions. <;1'4 is chosen for its particular elegance in representing rigid-body transforms, coupled with an equivalence relation between elements representing transforms more general than for ordinary homogeneous coordinates. The properties of the algebra germane to freeform motion design and improvement are given, and two distinct frameworks for freeform motion construction and modification are studied in detail.EThOS - Electronic Theses Online ServiceGBUnited Kingdo