Stereographic Projection

We give an explicit construction of any simply-connected superconformal surface $\phi\colon M^2\to \R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\colon M^2\to\R^4$. That $\phi$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of conjugate minimal surfaces that give rise to images of holomorphic curves

Study of the three treatises written by al-Sufi (X century) on the astrolabe (versions T, G, S) using 9 manuscripts. Study of the history of the astrolabe. Introduction to the theory of the stereographic projection for the construction of the astrolabe. Edition of the full Arabic text of chapters XV. 102-106 of G-version, which are concerned with the construction of the northern and southern astrolabe. English translation and commentary on these methods, enriched with additional figures. Full Arabic text and translation into English of S-version, which contains the quintessence of the knowledge on the astrolabe. Analysis of the methods of this version. Detailed comparison of S-version with T and G versions. This comparison leads to interesting conclusions for the evolution of ideas of al-Sufi and his course of research in Astronomy. Appendices: Glossaries of the technical terms and the astrolabe stars, titles of the chapters of T and G versions, correspondence between the chapters o...

Images can be transformed in many ways. Some ways, such as conformal mappings, allow the image to maintain its integrity and keep the original image identifiable. This paper analyzes different uses for these types of functions in the visual arts and utilizes neuroscience to underline their importance.

The astrolabe was the most widespread astronomical instrument from the Greek-Roman period to the end of the Middle Ages
and the beginning of the Renaissance. Among the most important treatises on the use of the astrolabe was those written by the
astronomer 'Abd al-Raḥman al-Ṣūfī (903-986AD).
In this paper, the outline of the 3 treatises of al-Ṣūfī on the astrolabe is presented, with a focus on the description of the
construction of the almucantars on a plate of the astrolabe, and the calculation of the distance between two cities using the
astrolabe. There is also a comparison between the 3 versions.
The work of al-Ṣūfī on the astrolabe inaugurated a new era in the study of the instrument. The preserved treatises on the use of
the astrolabe before al-Ṣūfī are short and elementary. Al-Ṣūfī tried to compile complete and detailed treatises, not only
including and systematizing the previous knowledge, but also correcting the errors of the former authors and expanding the
use of the astrolabe to thousands problems included in his treatises.

Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk $\mathbb D_1$), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from $N$ samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from $N$ samples is still possible and the accuracy of the approximation, which tends to be exact in the limit $N\to\infty$.

The digital revolution has increased the production and exchange of high-value documents between institutions, businesses and the general public. In order to secure these exchanges, it is essential to guarantee the authenticity, integrity and ownership of these documents. Digital watermarking is a possible solution to this challenge as it has already been used for copyright protection, source tracking and video authentication. It also provides integrity protection, which is useful for many types of documents (official documents, medical images). In this paper, we propose a new watermarking solution applicable to images and based on the hyperbolic geometry. Our new solution is based on existing work in the field of digital watermarking

It is highly desirable that an imaging system has a single effective viewpoint. Central catadioptric systems are imaging systems that use mirrors to enhance the field of view while keeping a unique center of projection. A general model for central catadioptric image formation has already been established. The present paper exploits this model to study the catadioptric projection of lines. The equations and geometric properties of general catadioptric line imaging are derived. We show that it is possible to determine the position of both the effective viewpoint and the absolute conic in the catadioptric image plane from the images of three lines. It is also proved that it is possible to identify the type of catadioptric system and the position of the line at infinity without further information. A methodology for central catadioptric system calibration is proposed. Reconstruction aspects are discussed. Experimental results are presented. All the results presented are original and completely new.

Joint French-Ethiopian field trips in 1995-1996 yield new geochronologic and paleomagnetic data, which significantly expand our knowledge of the recent magmatic and tectonic history of the Afar depression. Twenty-four new K-Ar ages range from 0.6 to 3.3 Ma. There is quite good agreement between magnetic polarities and Geomagnetic Polarity Timescale (GPTS). Eight age determinations with uncertainty less than 50 kyr

Given an embedded closed space curve with non-vanishing curvature, its self-linking number is defined as the linking number between the original curve and a curve pushed slightly off in the direction of its principal normals. We present an index formula for the self-linking number in terms of the writhe of a knot diagram of the curve and either (1) an