Singularities of helix surfaces in Euclidean 3-space

Wang, Yongqiao; Chang, Yuan; Liu, Haiming

doi:10.1016/j.geomphys.2020.103781

Cited by 6 publications

(1 citation statement)

References 13 publications

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“…Furthermore, singularity theory is a very useful tool to investigate the geometric properties of submanifolds in different ambient space (see, e.g., previous works [7][8][9][10][11][12][13] ). In this paper, as an application of the singularity theory, we mainly study the pseudo-spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3-space.…”

Section: Introductionmentioning

confidence: 99%

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

Wang

Yang

Chang

et al. 2022

Math Methods in App Sciences

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A surface with nonempty timelike, lightlike, and spacelike points in Minkowski 3‐space is a mixed type surface. The mixed type surface has a signature‐changing metric, and its lightlike points can be seen as singularities of such metric. In this paper, we study singular properties of pseudo‐spherical evolutes of lightlike loci on mixed type surfaces. We classify the generic singularities of pseudo‐spherical evolutes by the singularity theory. These singularities and the contact between lightlike loci and model submanifolds are closely associated. We also show the Legendrian duality among pseudo‐spherical evolutes, lightlike loci, and Darboux vectors. Moreover, the singularities of pseudo‐spherical evolutes are studied from the viewpoint of opening map‐germ. Finally, we give an example to demonstrate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

Wang

Yang

Chang

et al. 2022

Math Methods in App Sciences

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Generalized evolutes of planar curves

Wang

Chang

Liu

2021

Int. J. Geom. Methods Mod. Phys.

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The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

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Evolutoids and pedaloids of frontals on timelike surfaces

Wang,

Yang,

Chang

et al. 2023

Open Mathematics

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In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.

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Singularities of helix surfaces in Euclidean 3-space

Cited by 6 publications

References 13 publications

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

Generalized evolutes of planar curves

Evolutoids and pedaloids of frontals on timelike surfaces

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