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].

Distance-Genericity for Real Algebraic Hypersurfaces

1984 ◽  
Vol 36 (2) ◽  
pp. 374-384
Author(s):  
J. W. Bruce ◽  
C. G. Gibson
Keyword(s):  
Catastrophe Theory ◽  
Local Structure ◽  
Height Functions ◽  
Smooth Submanifold ◽  
Smooth Category ◽  
The Family ◽  
Algebraic Hypersurfaces ◽  
Focal Set ◽  
Topological Models

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

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).

The existence of sextactic points

1984 ◽  
Vol 96 (3) ◽  
pp. 433-436 ◽  
Author(s):  
D. L. Fidal
Keyword(s):  
Point Contact ◽  
Smooth Curve ◽  
Geometric Interpretation ◽  
Parameter Family ◽  
Existence Problem

Let M be a plane oval (a smooth curve without inflexions). In this note we show that a generic such M (where the precise assumptions will be stated later) has to have at least one sextactic point, that is a point p where the unique conic touching M at p with at least 5-point contact actually has 6-point contact. This existence problem came into prominence whilst [2] was being written. It was hoped to use the existence of sextactic points to show that the Morse transition on a 1-parameter family of focoids with signature 0 or 2 could not occur. The problem proved to be remarkably stubborn, however. Indeed, the geometric interpretation of sextactic points as given in § 3 was totally unexpected.

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.

Generic space curves and secants

1984 ◽  
Vol 98 (3-4) ◽  
pp. 281-289 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Local Structure ◽  
Natural Unit ◽  
Space Curves ◽  
Generic Curve

SynopsisIn this paper, we study the local structure of the secant mapping of a pair of disjoint curves. We show that for generic curves, the secant map and unit secant maps are locally stable. If we allow our curves to coincide, we can define anew unit secant map to be the natural unit tangent map near the diagonal. This is, for a generic curve, a locally stablemap away from the diagonal. Along the diagonal, it is locally stable as a ℤ2 symmetric germ (the ℤ2 symmetry originating with reflection in the diagonal).

Symmetry sets

1985 ◽  
Vol 101 (1-2) ◽  
pp. 163-186 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin ◽  
C. G. Gibson
Keyword(s):  
Smooth Manifold ◽  
Normal Forms ◽  
Shape Recognition ◽  
Uniqueness Result ◽  
Potential Functions ◽  
Plane Curves ◽  
Symmetry Set ◽  
Space Curves ◽  
The Family ◽  
Set Of Points

SynopsisFor a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).

Tooth Profile Generation of Point-Contact Involute Planetary Gear Drive with Small Teeth Difference

2013 ◽  
Vol 441 ◽  
pp. 561-567 ◽  
Author(s):  
Wen Cui Yi ◽  
Bao Ying Wang ◽  
Shu He
Keyword(s):  
Three Dimensional ◽  
Point Contact ◽  
Planetary Gear ◽  
Smooth Curve ◽  
Tooth Surface ◽  
Three Dimensional Model ◽  
Gear Drive ◽  
Tooth Surfaces ◽  
Meshing Characteristics ◽  
Internal Gear

A new kind of involute planetary gear drive with small teeth difference which meshes in point contact is put forward based on the kinematic method of gear geometry in this paper. The generation principle of tubular tooth surface is proposed according to a selected smooth curve attached to the planetary wheel tooth surface. The general equations of conjugate tooth surfaces are derived and its conjugated contact curve on the internal gear is also determined. Then the mathematical model is established. The parameter selection and meshing characteristics of point-contact involute planetary gear drive with small teeth difference are discussed respectively. The three dimensional model is finally established and its motion simulation is worked out. It lays the significant theoretical foundation and practical meaning for improving the gear transmission performance.

Geometry of projection-generic space curves

2009 ◽  
Vol 147 (1) ◽  
pp. 115-142
Author(s):  
C. T. C. WALL
Keyword(s):  
Local Structure ◽  
Double Point ◽  
Normal Forms ◽  
Blow Up ◽  
Complex Case ◽  
Plane Curves ◽  
Space Curves ◽  
Open Set ◽  
The Family ◽  
Flat Family

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

scholarly journals Morphological versus molecular delimitation of ciliate species: a case study of the family Clevelandellidae (Protista, Ciliophora, Armophorea)

2020 ◽  
Author(s):  
Lukáš Pecina ◽  
Peter Vďačný
Keyword(s):  
18S Rrna ◽  
18S Rrna Gene ◽  
Rrna Gene ◽  
Gene Sequences ◽  
Molecular Analyses ◽  
Ciliate Species ◽  
The Family ◽  
Geometrical Data ◽  
Geometrical Information

The endozoic ciliates of the family Clevelandellidae Kidder, 1938 typically inhabit the hindgut of wood-feeding panesthiine cockroaches. To assess the consistency of species delimitation in clevelandellids, we tested the utility of three sources of taxonomic data: morphometric measurements, cell geometrical information, and 18S rRNA gene sequences. The morphometric and geometrical data delimited the clevelandellid morphospecies consistently and unambiguously. However, only Paraclevelandia brevis Kidder, 1937 represented a homogenous taxon in both morphological and molecular analyses; the morphospecies Clevelandella constricta (Kidder, 1937) and C. hastula (Kidder, 1937) contained two or three distinct, more or less closely related genotypes each; and the genetic homogeneity of the morphospecies C. panesthiae (Kidder, 1937) and C. parapanesthiae (Kidder, 1937) was not corroborated by the 18S rRNA gene sequences at all. Moreover, the 18S rRNA gene phylogenies suggested the C. panesthiae-like morphotype to be the ancestral phenotype from which all other clevelandellid morphotypes arose. The only exception was the C. constricta-like morphotype, which very likely branched off before the diversification of the C. panesthiae-like progenitor. The present molecular analyses also suggested that a huge proportion of the clevelandellid diversity still waits to be discovered, since examination of only four panesthiine populations revealed 10 distinct clevelandellid genotypes/molecular species.

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