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- Title
Generalized evolutes of planar curves.
- Authors
Wang, Yongqiao; Chang, Yuan; Liu, Haiming
- Abstract
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.
- Subjects
PLANE curves; LOCUS (Mathematics); CURVATURE; IMMERSIONS (Mathematics)
- Publication
International Journal of Geometric Methods in Modern Physics, 2021, Vol 18, Issue 14, p1
- ISSN
0219-8878
- Publication type
Article
- DOI
10.1142/S0219887821502224