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Non-parabolic Spatial Hybrid Framed Curves and Their Applications in the Spatial Hybrid Number Space

Kaixin Yao

TL;DR

Addresses the extension of framed-curve geometry to non-parabolic spatial hybrid numbers in the four-dimensional, non-commutative space $\mathbb{H}^p$, including singularities. Proposes a construction: $(\gamma,\bm\nu_1,\bm\nu_2)$ with curvature $(l,m,n,\alpha)$, and proves an existence/uniqueness theorem up to congruence under $\mathcal{G}$. Develops the associated generated curves—evolutes, involutes, pedal, and contrapedal—via distance-squared functions and provides explicit Frenet-type relations and identities. Demonstrates the framework through a detailed explicit example and discusses implications for geometric modeling and trajectory analysis in hybrid-number settings.

Abstract

In this paper, we define non-parabolic spatial hybrid framed curves in the spatial hybrid number space, which may have singularities, and prove the existence and uniqueness theorem for non-parabolic spatial hybrid framed curves. As appliciations, we define evolutes, involutes, pedal and contrapedal curves of non-parabolic spatial hybrid framed curves and discuss their relations.

Non-parabolic Spatial Hybrid Framed Curves and Their Applications in the Spatial Hybrid Number Space

TL;DR

Addresses the extension of framed-curve geometry to non-parabolic spatial hybrid numbers in the four-dimensional, non-commutative space , including singularities. Proposes a construction: with curvature , and proves an existence/uniqueness theorem up to congruence under . Develops the associated generated curves—evolutes, involutes, pedal, and contrapedal—via distance-squared functions and provides explicit Frenet-type relations and identities. Demonstrates the framework through a detailed explicit example and discusses implications for geometric modeling and trajectory analysis in hybrid-number settings.

Abstract

In this paper, we define non-parabolic spatial hybrid framed curves in the spatial hybrid number space, which may have singularities, and prove the existence and uniqueness theorem for non-parabolic spatial hybrid framed curves. As appliciations, we define evolutes, involutes, pedal and contrapedal curves of non-parabolic spatial hybrid framed curves and discuss their relations.
Paper Structure (5 sections, 7 theorems, 53 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-parabolic spatial hybrid framed curves
  4. Curves generated by non-parabolic spatial hybrid framed curves
  5. Example

Key Result

Proposition 2.1

Take $\bm H_k = b_k \bm i + c_k \bm \varepsilon + d_k \bm h \in \mathbb H^p, ~ k = 1,2,3$, then

Figures (2)

  • Figure 1: A non-parabolic spatial hybrid framed base curve (black) with its evolute (blue) and involute (red).
  • Figure 2: A non-parabolic spatial hybrid framed base curve (black) with its pedal curve (green) and contrapedal curve (magenta).

Theorems & Definitions (23)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Theorem 3.6: Existence and uniqueness theorem
  • ...and 13 more