Differential geometry of space curves: forgotten chapters (Q6638642)

This paper constitutes a very nice introduction to some classical kinds of curves and surfaces in differential geometry and further explorations in an appealing and self-contained way. First, the definition and properties of classical planar evolutes and involutes of a planar curve are recalled. Later, the corresponding extensions of these two concepts in \(\mathbb{R}^3\) are addressed, giving rise to a very detailed study of their properties and their relations with other classical concepts such as developable surfaces, regression edges, osculating circles and osculating spheres.\N\NIn this setting, the evolute is defined as the locus of centers of osculating spheres. Equivalently, it is the curve whose osculating planes are normal planes of the given curve. A modification of the construction of these curves is then considered, in which normal planes of the initial curve are replaced by rectifying planes. This leads to the concept that the authors call pseudo-evolute (and the converse operation is called a pseudo-involute).\N\NIn the plane, involutes are constructed by wrapping an unstretchable string around a curve. The analogous construction in space provides another definition of an involute (and thus of its converse, the evolute), which the authors call Monge involute and Monge evolute. Some properties are studied and, finally, it is shown that the pseudo-evolute of a curve is the locus of centers of the osculating spheres of its Monge evolute.