Curvature for everyone

The first derivative is about approximation by a linear object. Curvature is a measure of the distance of a smooth nonlinear object from being linear, straight, flat, . . . . Curvature has been studied at least as long ago as the ancient Greeks, and it remains a very tumultuous research area today under the names of differential geometry and its applications. While having been a core subject of study in most university mathematics curricula for many decades, its overly precise language and humongous formulas have always been a major obstacle to making curvature and differential geometry widely accessible. The advent of computer algebra systems (CAS) is completely changing the playing field: The fundamental concepts and insights related to curvature are becoming accessible to a much larger part of the populace. At the same time, CAS allow one to shift from mostly very abstract theories with extremely few computable, nontrivial examples to lots of interactive experimentation largely based on computer visualization, but also on computer algebra. The opportunities for undergraduate research projects appear endless at this time! This article is to present some background material surveying selected basic notions and also objectives. It accompanies a live presentation whose heart are interactive implementations into the CAS MAPLE of some of the most intriguing topics of elementary differential geometry. The main objective is to demonstrate how much technology has changed the access to fundamental notions of curvature and closely related topics.