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From Normal Surfaces to Normal Curves to Geodesics on Surfaces

Eli Appleboim
2017 Axioms  
This paper gives a study of a two dimensional version of the theory of 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).  ... 
doi:10.3390/axioms6030026 fatcat:xeiptycvzzebrhwnt3kp63ruia

Normal Congruences Determined by Centers of Geodesic Curvature

Frederick Wahn Beal
1913 American Journal of Mathematics  
In that proof t was measured 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.  ... 
doi:10.2307/2370202 fatcat:5jkwmhm5svcgpipunblapcz54i

What is an Almost Normal Surface [article]

Joel Hass
2012 arXiv   pre-print
We explain how almost 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.  ... 
arXiv:1208.0568v2 fatcat:eej2d5j4cfh5dopmfhh5uckiv4

Darboux Iso-Geodesic Special Curve in Euclidean Space

M. M. Wageeda, E. M. Solouma, M. Bary
2019 Modern Applied Science  
In this paper, by using Darboux frame we scrutinize the issues of reconstructing 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.  ... 
doi:10.5539/mas.v13n9p98 fatcat:zjhzlai3wvf73pdc7w4c7zovim

Recurrent Geodesics on a Surface of Negative Curvature

Harold Marston Morse
1921 Transactions of the American Mathematical Society  
Each of these funnels will be cut off 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  ... 
doi:10.2307/1988844 fatcat:5ksihjvp4bb37kkr54elhgqide

Recurrent geodesics on a surface of negative curvature

Harold Marston Morse
1921 Transactions of the American Mathematical Society  
Each of these funnels will be cut off 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  ... 
doi:10.1090/s0002-9947-1921-1501161-8 fatcat:nsjbjdxgqnecjacnypcxhvaw6i

Total geodesic curvature and geodesic torsion

J. K. Whittemore
1923 Bulletin of the American Mathematical Society  
Let r be any 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  ... 
doi:10.1090/s0002-9904-1923-03659-8 fatcat:plfekzuusze5xabcbvw23qlziu

On meusnier theorem for parallel surfaces

SAVCI Ümit Ziya; GÖRGÜLÜ
2017 Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics  
In this paper, the 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 .  ... 
doi:10.1501/commua1_0000000788 fatcat:oqbqtgj6ebdm5abqnhlcf2tqzq

Polar Isodistance Curves on Parametric Surfaces [chapter]

J. Puig-Pey, A. Gálvez, A. Iglesias
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.  ... 
doi:10.1007/3-540-46080-2_17 fatcat:fbn63ksblzeaxisdoav353lh7u

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

Fathi M. Hamdoon, A.K. Omran
2016 Korean Journal of Mathematics  
We obtained some conditions 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  ... 
doi:10.11568/kjm.2016.24.4.613 fatcat:zttjldprdbge7ifzidxcimo7ha

Golab's Theorem

Curtis M. Fulton
1966 Proceedings of the American Mathematical Society  
(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.  ...  (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.  ... 
doi:10.2307/2036099 fatcat:f4cataibojdp7av3hanbakom4y

Timelike tubes with Darboux frame in Minkowski 3-space
English

Kızıltuğ Sezai, Yaylı Yusuf
2013 International Journal of Physical Sciences  
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 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.  ... 
doi:10.5897/ijps12.602 fatcat:dewi4izdqrdthgol7sc7er3bya

Existence conditions for Coons patches interpolating geodesic boundary curves

R.T. Farouki, N. Szafran, L. Biard
2009 Computer Aided Geometric Design  
boundary 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.  ... 
doi:10.1016/j.cagd.2009.01.003 fatcat:phssunan4rcajfqvwfyhed3krm

Golab's theorem

Curtis M. Fulton
1966 Proceedings of the American Mathematical Society  
(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.  ...  (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.  ... 
doi:10.1090/s0002-9939-1966-0198354-7 fatcat:xysoguizlvafrf364pvprgmegu

Congruences Associated with a One-Parameter Family of Curves

Ralph Dennison Beetle
1915 American Journal of Mathematics  
If S1 is the other focal 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 ?  ... 
doi:10.2307/2370397 fatcat:6etswd3xcbdmleofkog4sk2voq
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