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The behaviour of moving points on curves with rotating frames

Dong Han

Abstract

In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation. Conversely, it can also prove that a curve can be determined by the two motions of its moving point, namely, linear motion and rotation. Thus, we obtain a new binary mathematical formation mechanism for curves based on the aforementioned two motions. Finally, we apply this rotating frame method to the study of the behaviour of moving points on ellipses.

The behaviour of moving points on curves with rotating frames

Abstract

In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation. Conversely, it can also prove that a curve can be determined by the two motions of its moving point, namely, linear motion and rotation. Thus, we obtain a new binary mathematical formation mechanism for curves based on the aforementioned two motions. Finally, we apply this rotating frame method to the study of the behaviour of moving points on ellipses.
Paper Structure (19 sections, 23 theorems, 179 equations, 1 table)

This paper contains 19 sections, 23 theorems, 179 equations, 1 table.

Table of Contents

  1. Introduction
  2. The behaviour of moving points on plane curves
  3. Plane curves with general rotating frames 1
  4. Parameter transformation
  5. Plane curves with general rotating frames 2
  6. Plane curves with local rotating frames
  7. Plane lines
  8. The behaviour of moving points on space curves
  9. Space curves with general rotating frames
  10. Space curves with local rotating frames
  11. Two space curves with symmetric rotating frames
  12. The behaviour of moving points on curves lying on surfaces
  13. Curves on surfaces with general rotating frames
  14. Curves on surfaces with local rotating frames
  15. Application
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $P$ be a moving point on (xxsecpla1), and let its parameter be $t$.

Theorems & Definitions (45)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 35 more