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Pedal and contrapedal curves of mixed-type Minkowski plane curves

Xin Zhao, Pengcheng Li

Abstract

Pedal and contrapedal curves are important study objects of plane curves. As for a mixed-type Minkowski plane curve, since the definitions of the pedal and contrapedal curves at lightlike points can not always be given, the investigation of them is difficult. We have done some research on the pedal curves of a mixed-type curve. In this paper, we discuss when the contrapedal curves of a mixed-type curve exist and give the definition of them when they exist. Then, we study when the contrapedal curves of the mixed-type curve have singular points. Meanwhile, we consider the types of the points on the contrapedal curves. Moreover, we investigate the relationship between the pedal and contrapedal curves of a mixed-type curve, as well as the relationship among them and the evolute of the base curve.

Pedal and contrapedal curves of mixed-type Minkowski plane curves

Abstract

Pedal and contrapedal curves are important study objects of plane curves. As for a mixed-type Minkowski plane curve, since the definitions of the pedal and contrapedal curves at lightlike points can not always be given, the investigation of them is difficult. We have done some research on the pedal curves of a mixed-type curve. In this paper, we discuss when the contrapedal curves of a mixed-type curve exist and give the definition of them when they exist. Then, we study when the contrapedal curves of the mixed-type curve have singular points. Meanwhile, we consider the types of the points on the contrapedal curves. Moreover, we investigate the relationship between the pedal and contrapedal curves of a mixed-type curve, as well as the relationship among them and the evolute of the base curve.
Paper Structure (4 sections, 5 theorems, 55 equations, 5 figures)

This paper contains 4 sections, 5 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contrapedal curves of the mixed-type curves
  4. Relationships among pedal curves, contrapedal curves and evolutes of mixed-type curves

Key Result

Theorem 1

Let $\gamma:I\rightarrow\mathbb{R}_1^2$ be a regular mixed-type curve, $\boldsymbol{Q}$ be a point in $\mathbb{R}_1^2$ and $CPe(\gamma):I\rightarrow\mathbb{R}_1^2$ be the contrapedal curve of $\gamma$. Then, $(1)$ if $\gamma(t_0)$ is a non-lightlike point, then $CPe(\gamma)(t_0)$ is a singular point and $(2)$ if $\gamma(t_0)$ is a lightlike point, and $\boldsymbol{Q}$ is consistent with $\gamma(t

Figures (5)

  • Figure 1: The mixed-type curve (blue) and its contrapedal curves.
  • Figure 2: The mixed-type curve (blue) and its contrapedal curves.
  • Figure 3: The mixed-type curve (blue) and its contrapedal curves.
  • Figure 4: The mixed-type curve (blue), its pedal curve and contrapedal curve.
  • Figure 5: The mixed-type curve (blue) and its pedal curve and contrapedal curve.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Example 1
  • Example 2
  • Example 3
  • ...and 8 more