scholarly journals Canal surfaces with generalized 1-type Gauss map

2021 ◽  
pp. 199-211
Author(s):  
Jinhua Qian ◽  
Mengfei Su ◽  
Young Ho Kim
Keyword(s):  
Gauss Map ◽  
Canal Surfaces

Classifications of Canal Surfaces with L1-Pointwise 1-Type Gauss Map

2015 ◽  
Vol 83 (1) ◽  
pp. 145-155 ◽  
Author(s):  
Jinhua Qian ◽  
Young Ho Kim
Keyword(s):  
Gauss Map ◽  
Canal Surfaces

scholarly journals Classifications of Canal Surfaces with the Gauss Maps in Minkowski 3-Space

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1453
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Xueshan Fu ◽  
Young Ho Kim
Keyword(s):  
Minimal Surface ◽  
Gauss Map ◽  
Canal Surface ◽  
Surface Of Revolution ◽  
Canal Surfaces ◽  
Gauss Maps

In this work, we study the canal surfaces foliated by pseudo spheres S12 along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, the canal surface with proper pointwise 1-type Gauss map of the first kind if and only if it is a part of a minimal surface of revolution.

scholarly journals Surfaces of Revolution and Canal Surfaces with Generalized Cheng–Yau 1-Type Gauss Maps

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1728
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Xueqian Tian ◽  
Young Ho Kim
Keyword(s):  
Gauss Map ◽  
Profile Curve ◽  
Surfaces Of Revolution ◽  
Canal Surface ◽  
Open Part ◽  
Canal Surfaces ◽  
Gauss Maps ◽  
Unit Speed ◽  
Euclidean Three Space ◽  
Three Space

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

Blending Cylinders and Cones using Canal Surfaces

2000 ◽  
Vol 5 ◽  
pp. 77-89 ◽  
Author(s):  
M. Kazakevičiūtė ◽  
R. Krasauskas
Keyword(s):  
Canal Surface ◽  
Variable Radius ◽  
Rolling Ball ◽  
Canal Surfaces ◽  
Rational Parameterization

There is reviewed the construction of a rational blending surface between cylinders and cones in some interlocation cases. This surface is constructed as a patch of rolling ball envelope, i.e. as a patch of tangent canal surface of rational-variable radius. This construction defines rational parameterization of a blending surface. The constructed surface is Laguerre invariant.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

On the geometry of the Gauss map of conformal foliations by lines

2004 ◽  
Vol 136 (1) ◽  
pp. 247-255
Author(s):  
JEAN-MARIE BUREL ◽  
SIGMUNDUR GUDMUNDSSON
Keyword(s):  
Gauss Map

Ruled submanifolds with finite type Gauss map

1994 ◽  
Vol 49 (1-2) ◽  
pp. 42-45 ◽  
Author(s):  
Christos Baikoussis
Keyword(s):  
Finite Type ◽  
Gauss Map
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