The four basic geometric objects of points, point pairs, circles and spheres correspond to outer product null-spaces constructed by conformal points in conformal geometric algebra. Wedging with the infinity point we get four flat objects: finite-infinity point pairs, lines, planes and the whole 5D space. We show that all these basic geometric objects have the same algebraic structure. We then develop a single general algebraic method to quantify and interpolate the relative pose of the eight different classes of objects. As an explicit example we finally apply our framework to the conformation of organic macromolecules. Mathematics Subject Classification (2000). Primary 15A66; Secondary 62M45, 62H11, 92C40, 68T40.

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Although it is clear that rowers have a large muscle mass, their distribution of muscle mass and which of the main motions in rowing mediates muscle hypertrophy in each body part are unclear. We examine the relationships between partial motion power in rowing and muscle cross-sectional area of the thigh, lower back, and upper arms. Sixty young rowers (39 males and 21 females) participated in the study. Joint positions and forces were measured by video cameras and rowing ergometer software, respectively. One-dimensional motion analysis was performed to calculate the power of leg drive, trunk swing, and arm pull motions. Muscle cross-sectional areas were measured using magnetic resonance imaging. Multiple regression analyses were carried out to determine the association of different muscle cross-sectional areas with partial motion power. The anterior thigh best explained the power demonstrated by leg drive (r2 = 0.508), the posterior thigh and lower back combined best explained the power demonstrated by the trunk swing (r2 = 0.493), and the elbow extensors best explained the power demonstrated by the arm pull (r2 = 0.195). Other correlations, such as arm muscles with leg drive power (r2 = 0.424) and anterior thigh with trunk swing power (r2 = 0.33 5), were also significant. All muscle cross-sectional areas were associated with rowing performance either through the production of power or by transmitting work. The results imply that rowing motion requires a well-balanced distribution of muscle mass throughout the body.

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

Fuzzy models describe nonlinear input-output relationships with linguistic fuzzy rules. A hierarchical fuzzy modeling is promising for identification of fuzzy models of target systems that have many input variables. In the identification, (1) determination of a hierarchical structure of submodels, (2) selection of input variables of each submodel, (3) division of input and output space, (4) tuning of membership functions, and (5) determination of fuzzy inference method are carried out. This article presents a hierarchical fuzzy modeling method with an uneven division method of input space of each submodel. For selecting input variables of submodels, the multiple objective genetic algorithm (MOGA) is utilized. MOGA finds multiple models with different input variables and different numbers of fuzzy rules as compromising solutions. A human designer can choose desirable ones from these candidates. The proposed method is applied to acquisition of fuzzy rules from cyclists' pedaling data. In spite of a small number of data, the obtained model was able to give detailed suggestions to each cyclist.