Modeling of (n,m)-Type Minkowski Pythagorean Hodograph Curves with Hopf Map and Applications
Year: 2024
Type: article
Abstract: In the present paper, regular spacelike spatial Minkowski Pythagorean hodograph (MPH) curves are characterized with rational rotation-minimizing frames (RRMFs). We define an Euler–Rodrigues frame (ERF) for MPH curves and by means of this concept, we reach the definition of MPH curves of type (n,m). Expressing the conditions provided by these curves in the form of a Minkowski–Hopf map that we define; it is aimed to establish a connection with the Lorentz force that occurs during the process of computer numerical control (CNC)-type sinker electronic discharge machines (EDMs). This approach is reinforced by split quaternion polynomials. We give conditions satisfied by MPH curves of low degree to be type (n,m) and construct illustrative examples. In five-axis CNC machines, rotation-minimizing frames are used for tool path planning, and in this way, unnecessary rotations in the tool frame are prevented and tool orientation is provided. Since we obtain MPH curves with RRMF using the ERF, finally we define the Fermi–Walker derivative and parallelism along MPH curves with respect to the ERF and give applications.
Source: Fractal and Fractional
Authors: Aziz Yazla, Muhammed Talat Sarıaydın
Institution: Selçuk University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,069
OpenAlex ID: W4404819311