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From Normal Surfaces to Normal Curves to Geodesics on Surfaces

2017
*
Axioms
*

This paper gives a study of a two dimensional version of the theory of

doi:10.3390/axioms6030026
fatcat:xeiptycvzzebrhwnt3kp63ruia
*normal**surfaces*; namely, a study o*normal**curves*and their relations with respect*to**geodesic**curves*. ... This study results with a nice discrete approximation of*geodesics*embedded in a triangulated orientable Riemannian*surface*. Experimental results of the two dimensional case are given as well. ... We will show that this method gives rise*to*a flow under which a*normal**curve*converges*to*a*geodesic*. Let C be a rectifiable*curve**on*a triangulated Riemannian*surface*(Σ, T , g). ...##
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Normal Congruences Determined by Centers of Geodesic Curvature

1913
*
American Journal of Mathematics
*

In that proof t was measured

doi:10.2307/2370202
fatcat:5jkwmhm5svcgpipunblapcz54i
*from*(-R2, 0, 0)*to*any*normal**surface*of the congruence and was given by dt R2dR2 If the distance is*to*be measured*from*P1,*to*t must be added the distance*from*(-R2, , 0 ... The distances*from*M*to*the two centers of*geodesic*curvature of the parametric*curves*are constant; hence the distance*from*either of these centers*to*P1 is constant. But in ? ... Any*surface*orthogonal*to*the congruence A B is generated by a point P at a constant distance*from*A*on*A B. ...##
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What is an Almost Normal Surface
[article]

2012
*
arXiv
*
pre-print

We explain how almost

arXiv:1208.0568v2
fatcat:eej2d5j4cfh5dopmfhh5uckiv4
*normal**surfaces*emerged naturally*from*the study of*geodesics*and minimal*surfaces*. ... Analogous patterns of*normal*and almost*normal**surfaces*led Rubinstein*to*an algorithm for recognizing the 3-sphere. ... For*curves**on*a*surface*, the analog of a*geodesic*then becomes a special type of*normal**curve*. ...##
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Darboux Iso-Geodesic Special Curve in Euclidean Space

2019
*
Modern Applied Science
*

In this paper, by using Darboux frame we scrutinize the issues of reconstructing

doi:10.5539/mas.v13n9p98
fatcat:zjhzlai3wvf73pdc7w4c7zovim
*surfaces*with given some unusual Smarandache*curves*in Euclidean 3-space, we make manifest the family of*surfaces*as a linear ... combination of the components of this frame and derive the necessary and sufficient conditions for coefficients*to*satisfy both the iso-*geodesic*and iso-parametric requirements. ... Acknowledgment We wish*to*express our profound thanks and appreciation*to*the editor and the referees for their comments and suggestions*to*improve the paper. ...##
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Recurrent Geodesics on a Surface of Negative Curvature

1921
*
Transactions of the American Mathematical Society
*

Each of these funnels will be cut off

doi:10.2307/1988844
fatcat:5ksihjvp4bb37kkr54elhgqide
*from*the rest of the*surface*along a simple closed*curve*. These*curves*will be taken sufficiently remote*on*the funnels*to*be entirely distinct*from**one*another. ... There is a*one**to**one*correspondence between the set of all geodesies lying wholly*on*S, and the set of all unending reduced*curves*, in which each*geodesic*corresponds*to*that unending reduced*curve*that ...##
###
Recurrent geodesics on a surface of negative curvature

1921
*
Transactions of the American Mathematical Society
*

Each of these funnels will be cut off

doi:10.1090/s0002-9947-1921-1501161-8
fatcat:nsjbjdxgqnecjacnypcxhvaw6i
*from*the rest of the*surface*along a simple closed*curve*. These*curves*will be taken sufficiently remote*on*the funnels*to*be entirely distinct*from**one*another. ... There is a*one**to**one*correspondence between the set of all geodesies lying wholly*on*S, and the set of all unending reduced*curves*, in which each*geodesic*corresponds*to*that unending reduced*curve*that ...##
###
Total geodesic curvature and geodesic torsion

1923
*
Bulletin of the American Mathematical Society
*

Let r be any

doi:10.1090/s0002-9904-1923-03659-8
fatcat:plfekzuusze5xabcbvw23qlziu
*curve*of a*surface*S and P be any point of T ; let cô be the angle measured*from*the positive direction of the principal*normal**to*T at P*to*the positive direction of the*normal**to*S at P, ... If*on*the other hand any*curve*is given, a "*surface*band," that is, the*curve*and the*normal**to*a*surface*at each point of the*curve*, is determined by a solution ü of this equation such that the given ... Since the spherical representation of a minimal*surface*is conformai, J \\r g = db V-K sin 2\p for any*curve**on*such a*surface*, where \f/ is the angle of the tangent*to*the*curve*and the tangent*to*either ...##
###
On meusnier theorem for parallel surfaces

2017
*
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
*

In this paper, the

doi:10.1501/commua1_0000000788
fatcat:oqbqtgj6ebdm5abqnhlcf2tqzq
*geodesic*curvature, the*normal*curvature, the*geodesic*torsion and the curvature of the image*curve**on*a parallel*surface*of a given*curve**on*a*surface*are obtained. ... Moreover, Meusnier theorem for parallel*surfaces*are discussed. ... A*surface*M r whose points are at a constant distance along the*normal**from*another*surface*M is said*to*be parallel*to*M . ...##
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Polar Isodistance Curves on Parametric Surfaces
[chapter]

2002
*
Lecture Notes in Computer Science
*

*To*this aim, two different sets of characteristic

*curves*are considered: the

*normal*section

*curves*and the

*geodesic*

*curves*. ... In this paper, a new method for interrogation of parametric

*surfaces*is introduced. The basic idea is

*to*consider the distance measured

*on*certain

*curves*

*on*a

*surface*as an interrogation tool. ... Firstly, we introduce two new families of characteristic

*curves*

*on*a

*surface*: the

*normal*section

*curves*and the

*geodesic*

*curves*. ...

##
###
STUDYING ON A SKEW RULED SURFACE BY USING THE GEODESIC FRENET TRIHEDRON OF ITS GENERATOR

2016
*
Korean Journal of Mathematics
*

We obtained some conditions

doi:10.11568/kjm.2016.24.4.613
fatcat:zttjldprdbge7ifzidxcimo7ha
*on*this*surface**to*ensure that this ruled*surface*is flat, II-flat, minimal, II-minimal and Weingarten*surface*. ... Moreover, the parametric equations of asymptotic and*geodesic*lines*on*this ruled*surface*are determined and illustrated through example using the program of mathematica. ... Studying*on*a skew ruled*surface*by using the*geodesic*Frenet trihedron 625 asymptotic line ...##
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Golab's Theorem

1966
*
Proceedings of the American Mathematical Society
*

(I) A

doi:10.2307/2036099
fatcat:f4cataibojdp7av3hanbakom4y
*curve**on*a*surface*of class Cl is called Pstraight if the tangent planes*to*the*surface*along the*curve*remain always perpendicular*to*a fixed direction. ... (II) A*curve**on*a*surface*of class C1 is called P-plane if the tangent planes*to*the*surface*along the*curve*are all parallel*to*a fixed direction. We wish*to*prove the following Theorem. ... (I) A*curve**on*a*surface*of class Cl is called Pstraight if the tangent planes*to*the*surface*along the*curve*remain always perpendicular*to*a fixed direction. ...##
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Timelike tubes with Darboux frame in Minkowski 3-space

English

2013
*
International Journal of Physical Sciences
*

English

Subsequently, we compute the Gaussian curvature, the mean curvature, and the second Gaussian curvature of timelike tube with Darboux frame and obtained some characterizations for special

doi:10.5897/ijps12.602
fatcat:dewi4izdqrdthgol7sc7er3bya
*curves**on*this ... timelike tube around a spacelike*curve*with timelike binormal. ... ACKNOWLEDGEMENT The authors would like*to*thank the referee for his/her valuable suggestions which improved the first version of the paper. ...##
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Existence conditions for Coons patches interpolating geodesic boundary curves

2009
*
Computer Aided Geometric Design
*

boundary

doi:10.1016/j.cagd.2009.01.003
fatcat:phssunan4rcajfqvwfyhed3krm
*curves*correspond*to**geodesics*of the*surface*. ... The possibility of constructing such a*surface*patch is shown*to*depend*on*the given boundary*curves*satisfying two types of consistency constraints. ... Acknowledgements This work has been accomplished during the visit of the third author*to*the Department of Mechanical and Aeronautical Engineering, University of California, Davis. ...##
###
Golab's theorem

1966
*
Proceedings of the American Mathematical Society
*

(I) A

doi:10.1090/s0002-9939-1966-0198354-7
fatcat:xysoguizlvafrf364pvprgmegu
*curve**on*a*surface*of class Cl is called Pstraight if the tangent planes*to*the*surface*along the*curve*remain always perpendicular*to*a fixed direction. ... (II) A*curve**on*a*surface*of class C1 is called P-plane if the tangent planes*to*the*surface*along the*curve*are all parallel*to*a fixed direction. We wish*to*prove the following Theorem. ... (I) A*curve**on*a*surface*of class Cl is called Pstraight if the tangent planes*to*the*surface*along the*curve*remain always perpendicular*to*a fixed direction. ...##
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Congruences Associated with a One-Parameter Family of Curves

1915
*
American Journal of Mathematics
*

If S1 is the other focal

doi:10.2307/2370397
fatcat:6etswd3xcbdmleofkog4sk2voq
*surface*of the*normal*congruence of tangents*to*the*geodesic*systein, the*curves*u=const.*on*S are*geodesics*and the*curves*v=const. are their conjugates.* Hence, by Theorem 3, ... Conversely,*curves**on*a*surface*, which lie also*on*spheres orthogonal*to*the*surface*, are lines of curvature of constant*geodesic*curvature. ... 7 lead*to*another characteristic property of 0-*surfaces*. When t=0, (32) becomes r If, furthermore, S is an 0-*surface*with the*curves*v =const.*geodesic*parallels, we have by ? ...
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