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Multiple View Geometry in Computer Vision.

Stabilization of constraints and integrals of motion in dynamical systems

A coral reef represents the net accumulation of calcium carbonate (CaCO3) produced by corals and other calcifying organisms. If calcification declines, then reef-building capacity also declines. Coral reef calcification depends on the saturation state of the carbonate mineral aragonite of surface waters. By the middle of the next century, an increased concentration of carbon dioxide will decrease the aragonite saturation state in the tropics by 30 percent and biogenic aragonite precipitation by 14 to 30 percent. Coral reefs are particularly threatened, because reef-building organisms secrete metastable forms of CaCO3, but the biogeochemical consequences on other calcifying marine ecosystems may be equally severe.

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Geochemical Consequences of Increased Atmospheric Carbon Dioxide on Coral Reefs

Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.

Finite Unitary Reflection Groups

HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such

Kalman Filtering for Spacecraft Attitude Estimation

Kaum jemals wird tin Werk eines Historikers einen so starken Reiz tiben und so tiefe Einblicke in das Wesen der Geschichte offnen wie Gedanken und Erinnerungen eines groBen Staatsmannes, welcher selbst ein langes Leben hindurch an fUhrender Stelle in die Geschicke der Welt eingegriffen hat und eine tiberlegene geistige Per sonlichkeit mit der Kraft ktinstlerischer schriftstellerischer Gestaltung verbindet. Solchc Werke, schon fUr die politische Geschichte eine kostbare Seltenheit, sind fiir die Geschichte der exakten Wissenschaften bis her wohl kaum geschrieben worden. Urn so notwendiger erschien es, als Felix Klein vor Jahresfrist starb, mit der Herausgabe seiner Vor lesungen zur Geschichte der Mathematik und mathematischen Physik des 19. Jahrhunderts nicht zu zogern. Diese Vorlesungen sind die reife Frucht eines reichen Lebens in mitten der wissenschaftlichen Ereignisse, der Ausdruck tiberlegener Weisheit und tiefen historischen Sinnes, einer hohen menschlichen Kultur und einer meisterhaften Gestaltungskraft; sie werden sicherlich auf aIle Mathematiker und Physiker und weit tiber diesen Kreis hin aus eine groBe Wirkung austiben. In einer Zeit, wo der Blick der Menschen auch in der Wissenschaft allzusehr am Gegenwartigen hangt und das Einzelne in unnatiirlicher VergroBerung und iiber triebener Bedeutung gegentiber dem Ganzen zu betrachten pflegt, kann das Kleinsche Werk vielen die Augen wieder offnen fUr die Zusammenhange und Entwicklungslinien unserer Wissenschaft im GroBen.

Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert

THIS book continues the translation of Klein's “Elementar Mathematik” which Messrs. Hedrick and Noble began with their translation of the volume on arithmetic, algebra, and analysis. The volume under review is that devoted to geometry. It cannot be claimed that the translation is perfect; some sentences are English only in the sense that English words are used, but in construction they are essentially German, and there are some eccentricities in spelling, for example, “parellelopiped”. There are also some slips which may irritate the reader: for example, on p. 180 a footnote reads, “In the Sixth Memoir on Quantities [sic], already cited (p. 145),” and it appears that the reference is to Cayley's memoir on quantics which is correctly quoted on p. 134, not p. 145.Elementary Mathematics from an Advanced Standpoint GeometryFelix Klein. Translated from the third German edition by E. R. Hedrick and Prof. C. A. Noble. Pp. ix + 214. (London: Macmillan and Co., Ltd., 1939.) 15s. net.

Felix Klein

Papers

Vorlesungen uber die Entwicklung der Mathematik im 19. Jahrhundert

Elementary mathematics from an advanced standpoint

Vergleichende Betrachtungen über neuere geometrische Forschungen

Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade

Ueber die Transformation siebenter Ordnung der elliptischen Functionen