Special Issue on Spherical Mathematics and Statistics
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Special Issue on Spherical Mathematics and Statistics Helmut Schaeben
Received: 6 August 2010 / Accepted: 13 August 2010 / Published online: 4 September 2010 © International Association for Mathematical Geosciences 2010
Spherical mathematics and spherical statistics, respectively, often came in second with major achievements; there are numerous examples ranging from geometry and probability to approximation and computer aided geometric design. Spherical probability and statistics came into existence only in the fifties of the last century with the most influential paper by Ronald Aylmer Fisher, published in the Proceedings of the Royal Society in 1953, which was triggered by problems analyzing paleomagnetic data related to Wegener’s hypothesis of continental drift and eventually became instrumental to establishing the theory of plate tectonics. About fifty years later, spherical statistics was apparently completed by spherical regression and came to a preliminary standstill as it seems. As of today, there are three major textbooks on spherical statistics, and there are two better-known ones on circular statistics. There are few instances wherein spherical reasoning took the lead. The Funk transform (introduced by Paul Funk in Mathematische Annalen in 1913 and 1916, respectively), as compared to the Radon transform (presented by Johann Radon to the Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig in 1917), is a famous example. However, nowadays the term “Radon transform” is often used in a generic way to assign the mean values along the geodesics of the domain of definition to a function regardless of what this domain is. Many generalizations and specifications exist, most notably its discrete version known as the Hough transform (U.S. Patent 3069654, 1962). Often, the sphere in mind is a mathematical model of the Earth’s shape, but in crystallography and mineralogy, for instance, it is the sphere of poles, where pole points represent crystallographic directions or axes, that is, pairs of antipodally symH. Schaeben () Dept. of Geosciences, Geomathematics & Geoinformatics, BvCotta Str. 2, Freiberg 09596, Germany e-mail: [email protected]
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Math Geosci (2010) 42: 727–730
metric directions. In both cases, it is the two-dimensional sphere in three-dimensional space. The sphere allows only for orthogonal or special orthogonal transformations, that is, rotations and inversion. At first glance, it seems to be a rather poor structure. Things turn more interesting when rotations are considered in terms of unit quaternions (as introduced by Benjamin Olinde Rodrigues in Journal of Mathématiques in 1840, and William Rowan Hamilton in his Lectures on Quaternions in 1853) living on the unit three-dimensional sphere in four-dimensional space. For instance, despite the difference in dimensions, statistics of rotations and axes (which may be subject to rotations) are basically the same. Only in practical applications does the dimension matter numerically when it comes to computations involving normal
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