Application of instantaneous invariants to Cardan positions for two parameters complex plane motion

This paper aims to represent the Cardan position in the complex plane using Bottema’s instantaneous invariants for two parameters motions. The study provides unchanging and simple quantities that simplify the analysis of complicated mechanical systems by determining complex number representations of Bottema’s instantaneous invariants for Cardan motions with two parameters. The method involves illustrating invariants based on two parameters in the complex plane, defining cycloidal and Cardan motions of two parameters, and deriving the conditions for the Cardan positions. The key results are the necessary and sufficient conditions for the Cardan position for two-parameter motions in the complex plane obtained based on these instantaneous invariants.