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From Normal Surfaces to Normal Curves to Geodesics on Surfaces
2017
Axioms
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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
1913
American Journal of Mathematics
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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]
2012
arXiv
pre-print
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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
2019
Modern Applied Science
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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
1921
Transactions of the American Mathematical Society
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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
1921
Transactions of the American Mathematical Society
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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
1923
Bulletin of the American Mathematical Society
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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
2017
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
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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]
2002
Lecture Notes in Computer Science
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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
2016
Korean Journal of Mathematics
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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
1966
Proceedings of the American Mathematical Society
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(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
2013
International Journal of Physical Sciences
English
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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
2009
Computer Aided Geometric Design
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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
1966
Proceedings of the American Mathematical Society
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(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
1915
American Journal of Mathematics
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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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