On the planar development of classical developable surfaces

This paper presents a self-contained and explicit derivation of the planar development of regular developable surfaces, starting from the coefficients of the first fundamental form. Under the developability condition, the induced metric has a polynomial structure that allows one to construct explicitly a local isometry onto the plane. The resulting expressions show that the development is governed by a single angular function based on one parameter.

The approach provides a unified treatment of classical examples such as cylinders, cones, and ruled surfaces generated by tangents to space curves, and yields a direct characterization of geodesics as straight lines in the developed plane. The purpose of the paper is expository: no new results are claimed, and the emphasis is on making explicit and systematic several classical formulas that are often presented only implicitly or in scattered form in the literature.