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In this work, we are interested in the differential geometry of curves in thesimply isotropic and pseudo-isotropic 3-spaces, which are examples ofCayley-Klein geometries whose absolute figure is given by a plane at infinityand a degenerate quadric. Motivated by the success of rotation minimizing (RM)frames in Euclidean and Lorentzian geometries, here we show how to build RMframes in isotropic geometries and apply them in the study of isotropicspherical curves. Indeed, through a convenient manipulation of osculatingspheres described in terms of RM frames, we show that it is possible tocharacterize spherical curves via a linear equation involving the curvaturesthat dictate the RM frame motion. For the case of pseudo-isotropic space, wealso discuss on the distinct choices for the absolute figure in the frameworkof a Cayley-Klein geometry and prove that they are all equivalent approachesthrough the use of Lorentz numbers (a complex-like system where the square ofthe imaginary unit is $+1$). Finally, we also show the possibility of obtainingan isotropic RM frame by rotation of the Frenet frame through the use ofGalilean trigonometric functions and dual numbers (a complex-like system wherethe square of the imaginary unit vanishes).

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces