Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These resul...

This article develops new techniques to classify critical configurations for 3D scene reconstruction from images taken by unknown cameras. Generally, all information can be uniquely recovered if enough and image points are provided, but there certain cases where unique recovery is impossible; these called configurations. In this paper, we use an algebraic approach study the two projective We sh...

The performance of several methods for estimating local surface geometry (the principal frame plus the principal quadric) are examined by applying them to a suite of synthetic and real test data which have been corrupted by various amounts of additive Gaussian noise. Methods considered include nite diierences, a facet based approach, and quadric surface tting. The nonlinear quadric tting method...

− The concept of the reflective solar tower is based on inverting the path of the solar rays originating from a heliostat field to a solar receiver that can be placed on the ground. This system is based on the property of a reflective quadric surface to reflect each ray oriented to one of its foci to its second focus. Two shapes of surfaces can be used: one is concave − the ellipsoid, and the o...

One of the most interesting tasks of scene analysis is the reconstruction problem, which is the problem of finding a three-dimensional description of a scene from two or more projections. When the scene is assumed to be composed of man-made objects bounded by quadric surfaces, the reconstruction process is simplified if the border lines in each projection are given by means of straight-line seg...

In this paper, we propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curve...

Based on Lloyd iteration, we present a variational method for extracting general quadric surfaces from a 3D mesh surface. This work extends the previous variational methods that extract only planes or special types of quadrics, i.e., spheres and circular cylinders. Instead of using the exact L error metric, we use a new approximate L error metric to make our method more efficient for computing ...

1. C. B. Boyer, History of Analytic Geometry, Scripta Mathematica, New York, 1956. 2. J. L. Coolidge, The origin of analytic geometry, Osiris 1 (1936) 231–250, also available at www.jstor. org. 3. M. Ghomi and B. Solomon, Skew loops and quadric surfaces, Comment. Math. Helv. 4 (2002) 767–782. 4. J.-P. Sha and B. Solomon, No skew branes on non-degenerate hyperquadrics, Math. Zeit. 257 (2002) 225...

Making suitable generalizations of known results we prove some general facts about Gaussian maps. These facts are then used, in the second part of the article, to give a set of conditions that insure the surjectivity of Gaussian maps for curves on Enriques surfaces. To do this we also solve a problem of independent interest: a tetragonal curve of genus g ≥ 7 lying on an Enriques surface and gen...

Given a surface or scattered data points from a surface in 3-space, we show how to approximate the given data by a ruled surface in tensor product B-spline representation. This leads us to a general framework for approximation in line space by local mappings from the Klein quadric to Euclidean 4-space. The presented algorithm for approximation by ruled surfaces possesses applications in archite...