A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces.

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric...

We present a parallel approach for optimizing surface meshes by redistributing vertices on a feature-aware higher-order reconstruction of a triangulated surface. Our method is based on a novel extension of the fundamental quadric, called the medial quadric. This quadric helps solve some basic geometric problems, including detection of ridges and corners, computation of one-sided normals along r...

A technique for reconstructing a class of quadric surfaces from 3D data is presented. The technique is driven by a linear least-squaresbased fitting mechanism. Previously, such fitting was restricted to recovery of central quadrics; here, extension of that basic mechanism to allow recovery of one commonly-occurring class of non-central quadric, the elliptic paraboloids, is described. The extens...

We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split surface.

In the 3-dimensional Euclidean space E3, a quadric surface is either ruled or of one following two kinds z2=as2+bt2+c,abc≠0 z=a2s2+b2t2,a>0,b>0. present paper, we investigate these three surfaces whose Gauss map N satisfies property ΔIIN=ΛN, where Λ square symmetric matrix order 3, and ΔII denotes Laplace operator second fundamental form II surface. We prove that spheres with nonzero Λ, h...