Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

The following notation will be used: There will be a “parameter plane”, E2, with Cartesian coordinates (u,v) or (u1,u2) and a Euclidean space E3 with Cartesian coordinates (x,y,z) or (x1,x2,x3).KeywordsTangent VectorFundamental FormTangent PlanePlane CurveGeodesic LineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Ruled surfaces, also known as scrolls, are the surfaces built up by moving a straight line along a certain course in the 3D Euclidean space (E³). Many geometers have studied ruled surfaces for centuries to gain rational knowledge about how the surfaces in E³ are generated, and their findings are now essential in all monographs on differential geometry. In the area of optical scanning, scan fields produced by mirror scanning devices of different configurations are surfaces with a light ray as its generating element. This implies, within the accuracy of geometrical optics, that all scan fields are ruled surfaces by nature, and the theory of ruled surfaces in differential geometry provides a land with precious deposits for structural analysis of the scan fields.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim sigma-models corresponding to coset conformal field theories.

This chapter briefly reviews the fundamental concepts and theorems in algebraic topology [1], surface differential geometry [6], and surface Ricci flow [4, 7]. Detailed discussion on Ricci flow on general Riemannian manifolds can be found in [5]. Advanced topics on differential geometry related to Yamabe equations can be found in [9].

As a general reference to the topological material covered in this chapter, see Seifert-Threlfall: Lehrbuch der Topologie.

The aim of this study is to give spinor representation of successive curves in three-dimensional Euclidean space. In three dimensional Euclidean Space, the spinor representations of a curve with unit speed and a successor curve with the same arc length parameter as this curve has been studied. For this, first of all, the curve with unit speed and its successor curve have been corresponded to two different spinors. Then, using the relationships between these curves, the relationships between the spinors corresponding to these curves have been given. Therefore, geometric interpretations of these curves and corresponding spinors have been made. In addition, different spinor equations of the mates and derivatives of spinors have been examined and geometric interpretations of these spinor equations have been given. Then, spinor equations have been obtained in case the successive curves are helices. Consequently, two examples have been given.

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.

Boundary integration of polynomials over an arbitrary linear hexahedron in euclidean three-dimensional space

This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite.

A method for accurate registration on point clouds is presented in the paper. Manual alignment or the use of landmarks is avoided in the process of multi-view point clouds. Firstly, the differential geometric information is extracted from the point clouds. The extended Gaussian sphere and combination features are used to define the corresponding points of crude alignment. Secondly, the optimal algorithm，the point-to-point Iterated Closest Point, is applied to the accurate registration on point clouds. Thus, the complete point cloud can be obtained in the method.

Gauss as a geometer

Analysis of human back shape using surface curvatures

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.

Parallel curves have been used extensively in a variety of fields during the last decade, including architecture, computer graphics, aerospace, and medicine. Despite these applications, the theoretical development of parallel curves in differential geometry has been relatively limited. In this study, we investigate parallel curves using the q-frame in three-dimensional Euclidean space, rather than the traditional Frenet frame. We introduce the new expression “quasi-parallel curves” as a generalization of classical parallel curves. We also examine several geometric properties of these curves and provide rigorous proofs. In differential geometry, ruled surfaces are of great importance. They are defined by the movement of generators, which generate straight lines on the surface. Furthermore, a directrix (base curve) is any curve that intersects all generators (rulings). In this paper, we investigate a new class of quasi-ruled surfaces whose base curve is both the original curve and its quasi-parallel curve in R 3 . We describe the geometric characteristics of these surfaces and derive the first and second fundamental forms, as well as the Gaussian and mean curvatures. Specific types of quasi-ruled surfaces are analyzed, and conditions for their developability and minimality are established. Several quasi-ruled surfaces generated from the helix curve as a base curve and others generated from its quasi-parallel curve as a base curve are provided, and the parametrization of the resulting surfaces are determined. Finally, all generated surfaces, together with their quasi-parallel and original curves, are visualized using Mathematica 13.

There are two main differences between the theory of regular curves andregular surfaces in three-dimensional Euclidean spaces. On one hand, smooth curves are mappings (i.e.KeywordsVector FieldCovariant DerivativeFundamental FormTangent PlaneChristoffel SymbolThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.