Tangential developable

In the mathematical study of the differential geometry of surfaces, a tangential developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surface is also the envelope of the tangent planes to the curve.

With the exceptions of the plane, a cylinder, and a cone, every developable surface in three-dimensional Euclidean space is the tangential developable of a certain curve, the edge of regression. This curve is obtained by first developing the surface into the plane, and then considering the image in the plane of the generators of the ruling on the surface. The envelope of this family of lines is a plane curve whose inverse image under the development is the edge of regression. Intuitively, it is a curve along which the surface needs to be folded during the process of developing into the plane.

References

• Struik, Dirk Jan (1961), Lectures on Classical Differential Geometry, Addison-Wesley.
• Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, ISBN 978-0-8284-1087-8 {{citation}}: ISBN / Date incompatibility (help)
• Sabitov, I.Kh. (2001) [1994], "Developable surface", Encyclopedia of Mathematics, EMS Press
• Voitsekhovskii, M.I. (2001) [1994], "Edge of regression", Encyclopedia of Mathematics, EMS Press