Generalized evolutes of planar curves

2021 ◽  
Author(s):  
Yongqiao Wang ◽  
Yuan Chang ◽  
Haiming Liu
Keyword(s):  
Euclidean Plane ◽  
Singular Points ◽  
Moving Frames ◽  
Planar Curves ◽  
Fixed Direction ◽  
Spatial Curves

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.

Offset Curves and Surfaces With Continued Fractions

1999 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Continued Fractions ◽  
Free Form ◽  
New Approach ◽  
Planar Curves ◽  
Curves And Surfaces ◽  
Cad Cam ◽  
Offset Curves ◽  
Unified Algorithm ◽  
The Relationship ◽  
Spatial Curves

Abstract This paper provides a new approach to approximate the offset of free-form curves and surfaces with continued fractions. The properties of continued fractions are introduced. The convergence of the approximation is proved and the relationship between accuracy and approximate step is established. It shows that the approximation converges fast and reliably, the error of approximation is easily estimated and controlled, and a unified algorithm can be used to generate the rational offset of non-rational and rational planar curves, spatial curves, and surfaces in CAD/CAM industry.

scholarly journals On some Formulas about Volume and Surface Area

1953 ◽  
Vol 6 ◽  
pp. 109-117
Author(s):  
Minoru Kurita
Keyword(s):  
Surface Area ◽  
Euclidean Plane ◽  
Moving Frames ◽  
Integral Formulas

We prove in this paper some integral formulas about volume and surface area which are the extensions of the classical formulas such as Guldin-Pappus’s theorem about the solid of rotation and the surface of rotation and Holditch’s theorem about the area of the domains bounded by the loci of three points on a segment that moves on the euclidean plane. The formulas we prove are so elementary that they may be found in some literature, but the proofs here given are very simple by the use of moving frames and I assume that they are of some interest.

Discrete Invariant Curve Flows, Orthogonal Polynomials, and Moving Frame

2020 ◽  
Author(s):  
Bao Wang ◽  
Xiang-Ke Chang ◽  
Xing-Biao Hu ◽  
Shi-Hao Li
Keyword(s):  
Orthogonal Polynomials ◽  
Affine Space ◽  
Affine Plane ◽  
Euclidean Plane ◽  
Invariant Curve ◽  
Moving Frame ◽  
High Dimensional ◽  
Moving Frames ◽  
Curve Flows ◽  
Explicit Expressions

Abstract In this paper, an orthogonal polynomials-based (OPs-based) approach to generate discrete moving frames and invariants is developed. It is shown that OPs can provide explicit expressions for the discrete moving frame as well as the associated difference invariants, and this approach enables one to obtain the corresponding discrete invariant curve flows simultaneously. Several examples in the cases of centro-affine plane, pseudo-Euclidean plane, and high-dimensional centro-affine space are presented.

scholarly journals The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Wojciech Domitrz ◽  
Michał Zwierzyński
Keyword(s):  
Rotation Number ◽  
Optical Lattice ◽  
Quantum Particle ◽  
Singular Points ◽  
Planar Curves ◽  
Lattice Potential ◽  
Global Properties ◽  
Important Object ◽  
Cusp Singularities ◽  
Regular Closed

AbstractIn this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

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