Surfaces In Three-space Research Articles

Self-intersections of surfaces that contain two circles through each point

We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the topology of the surface.

Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

AbstractFejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

Higher solutions of Hitchin’s self-duality equations

AbstractSolutions of Hitchin’s self-duality equations correspond to special real sections of the Deligne–Hitchin moduli space—twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.

Deformation of tropical Hirzebruch surfaces and enumerative geometry

We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces Σ n \Sigma _n and Σ n + 2 \Sigma _{n+2} , obtained by deforming the first surface to the latter. Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces and tropical enumerative geometry on a tropical surface in three-space.

Planar surfaces in positive characteristic

A planar surface is a surface in three-space in which every tangent line has triple or higher contact with the surface at the point of tangency. We study properties of planar surfaces in positive characteristics, use that to bound the number of points of a planar surface over a finite field and give an application to Waring’s problem for polynomials.

Enumeration of tropes

Singularities of the plane sections of a general surface in three-space are well known, and counted; in particular, all plane sections are reduced. Fixing integers d, k ,w e give formulas for the degree of the locus of surfaces of degree d admitting a plane which is tangent along some curve of degree k.

Bifurcations of affine equidistants

The bifurcations of so-called affine equidistants for a surface in three-space are classified and described geometrically. An affine equidistant is formed by the points dividing in a given ratio the segment with the endpoints lying on a given surface provided that the tangent planes to the surface at these endpoints are parallel. The most interesting case corresponds to segments near parabolic lines. All singularities turn out to be stable and simple.

Pseudotriangulations from Surfaces and a Novel Type of Edge Flip

We prove that planar pseudotriangulations have realizations as polyhedral surfaces in three-space. Two main implications are presented. The spatial embedding leads to a novel flip operation that allows for a drastic reduction of flip distances, especially between (full) triangulations. Moreover, several key results for triangulations, like flipping to optimality, (constrained) Delaunayhood, and a convex polytope representation, are extended to pseudotriangulations in a natural way.

Fluctuation of Sectional Curvature for Closed Hypersurfaces

Liebmann proved in 1899 that the only closed surfaces in Euclidean three-space that have constant Gauss curvature are round spheres. Thus, if a closed surface in three-space is not a topological sphere, its Gauss curvature must fluctuate. We consider quantitative formulations of this fact, also in higher dimensions.

Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).

Algorithms for G 1 connection of multiple parametric bicubic NURBS surfaces

The objective of this paper is to introduce an innovative approach for constructing effective algorithms for removing gaps between parametric NURBS surfaces in three-space, while maintaining geometrical smoothness for the combined (or compound) surface. Similar to the degenerate case of tensor-product B-spline surfaces, if the underlying knot sequences along the connecting boundaries of two NURBS surfaces are proportional, then the parametric surfaces can be connected in a G1 fashion. This approach can be easily extended to connecting three or four parametric NURBS surfaces. We will demonstrate the feasibility of our approach by focusing on the C1 bicubic setting with knot sequences being equally-spaced and having double interior knots.

On Some Classical Methods to Improve Algebraic Curves and Surfaces

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.

Inhomogeneous surface diffusion for image filtering

Most of the recent work on inhomogeneous diffusion in image filtering focuses on diffusing the isotope curve. We present a less familiar approach to the development of inhomogeneous diffusion algorithms in which the image is regarded as a surface in three-space. The magnitude of the surface normal controls a diffusion that evolves the image surface at a speed proportional to its mean curvature. A discrete algorithm to implement this proposed diffusion is introduced, and we show experimentally that the algorithm develops singularities that reveal, preserve, and enhance the underlying signal. If the input signal is an isolated noisy edge, this leads to complete noise removal and enhancement without affecting the edge locality. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 450–456, 1997

Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms

We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke's Surface Evolver [Brakke 1992]. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.

Global inequalities for curves and surfaces in three-space

The aim of the paper is to give upper bounds for the total curvature of smooth curves and surfaces embedded in euclidean space, in terms of other global geometric characters; in particular, for a plane curve γ, we prove the inequality K(γ) < π(2 + f(γ)d(γ)/2), where d(γ) is the geometric degree of γ and f(γ) is the number of its inflection points. In the case of a surface S, a bound is given in terms of the genus g(S), the number of components of the parabolic points on S and the geometry of its apparent contour.

Shape and texture from serial contours

Useful shape descriptors for three-dimensional objects can be computed from serial (contour)sections reconstructed into surfaces in three-space. Fischler's algorithm for the caliper diameter of a planar region generalizes in an obvious way to give the major axes of an ellipsoid approximating the shape of a three-dimensional object. The orientation of the object within the three-dimensional sample space, or scene, is a byproduct of the calculation. The anisotropy of the scene can also be described. Fischler's algorithm, together with the classical shape descriptor of perimeter squared error (P2/A)area, can be used to decide the best approximating polygons or ellipses for calculating an object's volume by summing over its successive contours.

Geometrical class and degree for surfaces in three-space

Geometrical class and degree for surfaces in three-space

The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces

rigid, then the whole surface is rigid. It had been suspected that any (connected) polyhedral surface, convex or not, (with its triangular faces, say, held rigid) was rigid, but this has turned out to be false (see Connelly [7]). W e extend Cauchy’s theorem to show that any convex polyhedral surface, no matter how it is triangulated, is rigid. Note that vertices are allowed in the relative interior of the natural faces and edges. Here we say a triangulated surface in three-space is rigid if any continuous deformation -of the surface that keeps the distance fixed between any pair of points in each triangle (a part of the triangulation of the surface), keeps the distance fixed between any pair of points on the surface (and thus extends to an isometry of three-space). A natural face of a convex polyhedral surface is the two dimensional intersection of a support plane with the surface (see Fig. 3). Similarly natural edges and vertices are one- and zero-dimensional intersections, respectively. Thus in Cauchy’s theorem it is insisted that under a deformation of the surface the distance is fixed between any pair of points of a natural face and not just some triangle of a triangulation. Alexandrov [l] showed that if a convex polyhedral surface is triangulated with no vertices in the relative interior of a natural face (but they are allowed in natural edges), then the surface is rigid. This theorem is the basic result for the results of this paper. The author is grateful for a description of Alexandrov’s proof by Asimow and Roth [5]. Whiteley [ 151 also has a somewhat

Some remarks concerning surfaces in three-space

Some remarks concerning surfaces in three-space

A Geometrical and Highly Generalized Interpretation of Two-Element-Kinl One-Port Driving Functions

This paper presents in terms of a three-dimensional surface a technique for a visual as well as mathematical treatment of broad generality of the RL, RC and LC one-ports. An examination of the partial fraction terms of all the two-element series and parallel components which make up the canonic one-ports indicates that seven different types of terms are all that occur. The real and imaginary parts of all such functions for s = σ + jω may be represented geometrically by a surface in three-space. It is found that all the real and imaginary part functions are geometrically 1) a plane parallel or inclined to Aar the a, o plane; 2) the basic surface, given by 2 + 2' A > O, located in various positions with respect to the axes; or 3) a combination of the plane and basic surface. The driving function of any two-element-kind one-port is the sum of some combination of the real and imaginary part functions and so is represented by a sum of some combination of the canonic surfaces.