Related papers: Normal curves on a smooth immersed surface
We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to M\"obius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in…
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a…
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…
We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of…
In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow…
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding…
We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a $C^2$-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations…
In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an…
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality.
At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show…
Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately, current approaches are only available for the most basic geometries and fall short when the underlying…
We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality are computed by pencils on the ambient surface. From the geometrical view point, this means that the…
This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb{R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those…
We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on…
Mode surfaces are the generalization of degenerate curves and neutral surfaces, which constitute 3D symmetric tensor field topology. Efficient analysis and visualization of mode surfaces can provide additional insight into not only…
We give necessary conditions on the invariants (d,g) of a smooth, integral curve self-linked by a complete intersection of type (a,b) in projective three space. Similar conditions are given for s.t.c.i. curves with a multiplicity three…
We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not…