Construction and Analysis of Transcendental Curves by Means of Computer Geometry

A methodology is presented for the algorithmic construction and detailed differential-geometric analysis of a family of transcendental curves described by points of planetary mechanisms (hypocycloids, epicycloids, cycloids, ellipses). The methodology is implemented as a complex of universal programs within the Maple mathematical software environment, which allows combining precise symbolic calculations with powerful graphical modeling tools. The complex automates key stages of geometric analysis: from generating the parametric equations of the curve to calculating and visualizing differential characteristics such as the magnitude of the tangent vector (velocity), the vector of curvature (acceleration), and the radius of curvature ρ at any given point. Particular attention is paid to the graphical representation of the mechanisms for reproducing the curves in the form of two- and three-dimensional animated models that display the base points and their trajectories. The development is adapted for use in courses on mechanics, computer geometry, and engineering graphics, allowing for the generation of unique multivariate educational problems and contributing to the development of algorithmic and spatial thinking skills.