scholarly journals Evolutes and focal surfaces of framed immersions in the Euclidean space

2019 ◽  
Vol 150 (1) ◽  
pp. 497-516 ◽  
Author(s):  
Shun'ichi Honda ◽  
Masatomo Takahashi
Keyword(s):  
Euclidean Space ◽  
Smooth Curve ◽  
Singular Points ◽  
Moving Frame ◽  
Regular Space ◽  
Space Curves ◽  
Focal Surface

AbstractWe consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

scholarly journals Framed curves in the Euclidean space

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Shun’ichi Honda ◽  
Masatomo Takahashi
Keyword(s):  
Euclidean Space ◽  
Tangent Bundle ◽  
Moving Frame ◽  
Regular Curve ◽  
Smooth Functions ◽  
Independent Condition ◽  
Legendre Curve ◽  
Space Curves ◽  
Unit Tangent Bundle ◽  
Legendre Curves

AbstractA framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

scholarly journals Complete intersections primitive structures on space curves

2015 ◽  
Vol 26 (12) ◽  
pp. 1550104
Author(s):  
Philippe Ellia
Keyword(s):  
Complete Intersection ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Complete Intersections ◽  
Multiple Structure ◽  
Space Curves ◽  
Structure Formula ◽  
Primitive Structure

A multiple structure [Formula: see text] on a smooth curve [Formula: see text] is said to be primitive if [Formula: see text] is locally contained in a smooth surface. We give some numerical conditions for a curve [Formula: see text] to be a primitive set theoretical complete intersection (i.e. to have a primitive structure which is a complete intersection).

Local Parametrization of Space Curves at Singular Points

1992 ◽  
pp. 61-90 ◽  
Author(s):  
Maria E. Alonso ◽  
Teo Mora ◽  
Gianfranco Niesi ◽  
Mario Raimondo
Keyword(s):  
Singular Points ◽  
Space Curves ◽  
Local Parametrization

scholarly journals Generic isotopies of space curves

1987 ◽  
Vol 29 (1) ◽  
pp. 41-63 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin
Keyword(s):  
Point Contact ◽  
Smooth Curve ◽  
Singular Surfaces ◽  
Height Functions ◽  
Space Curves ◽  
Local Structures ◽  
The Family ◽  
Generic Curve ◽  
Geometrical Information ◽  
Focal Set

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

scholarly journals Space curves defined by curvature-torsion relations and associated helices

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4951-4966
Author(s):  
Sharief Deshmukh ◽  
Azeb Alghanemi ◽  
Rida Farouki
Keyword(s):  
Euclidean Space ◽  
Space Curves ◽  
General Polynomial ◽  
Slant Helix ◽  
Pythagorean Hodograph ◽  
Unit Speed ◽  
Circular Helix

The relationships between certain families of special curves, including the general helices, slant helices, rectifying curves, Salkowski curves, spherical curves, and centrodes, are analyzed. First, characterizations of proper slant helices and Salkowski curves are developed, and it is shown that, for any given proper slant helix with principal normal n, one may associate a unique general helix whose binormal b coincides with n. It is also shown that centrodes of Salkowski curves are proper slant helices. Moreover, with each unit-speed non-helical Frenet curve in the Euclidean space E3, one may associate a unique circular helix, and characterizations of the slant helices, rectifying curves, Salkowski curves, and spherical curves are presented in terms of their associated circular helices. Finally, these families of special curves are studied in the context of general polynomial/rational parameterizations, and it is observed that several of them are intimately related to the families of polynomial/rational Pythagorean-hodograph curves.

scholarly journals Developable surfaces in Euclidean space

1999 ◽  
Vol 66 (3) ◽  
pp. 388-402 ◽  
Author(s):  
Vitaly Ushakov
Keyword(s):  
Euclidean Space ◽  
Arbitrary Dimension ◽  
Main Tool ◽  
Moving Frame ◽  
Two Dimensional ◽  
Surfaces In Euclidean Space ◽  
Moving Frame Method ◽  
Frame Method ◽  
Euclidean Three Space ◽  
Three Space

AbstractThe classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

scholarly journals Tire Tracks and Integrable Curve Evolution

2018 ◽  
Vol 2020 (9) ◽  
pp. 2698-2768 ◽  
Author(s):  
Gil Bor ◽  
Mark Levi ◽  
Ron Perline ◽  
Sergei Tabachnikov
Keyword(s):  
Euclidean Space ◽  
Berry Phase ◽  
Three Dimensional ◽  
Rear Wheel ◽  
Dimensional Euclidean Space ◽  
First Order ◽  
Space Curves ◽  
Akns System ◽  
Special Case

Abstract We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation—the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the Ablowitz, Kaup, Newell, and Segur (AKNS) system and the vortex filament equation. We show that “bicycle correspondence” of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a three-dimensional bike of imaginary length. We show that a series of examples of “ambiguous” closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.

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