Talk:Developable surface

"what" "is" "the" "deal" "with" "all" "the" "quotation" "marks" —Preceding unsigned comment added by Monguin61 (talk • contribs) 05:11, 17 March 2009 (UTC)

In mathematics, a developable surface is a surface with zero Gaussian curvature.

Wondering if having everywhere zero curvature implies that the surface is developable? If so is their a proof of the result? --Salix alba (talk) 19:55, 27 February 2007 (UTC)

By definition, surface is developable if and only if it has zero Gaussian curvature (sum of angles of any triangle on that surface always equals to 180°). Admiral Norton ^(talk) 20:57, 19 February 2008 (UTC)

Complete classification?

I've just removed the following reference from the article: Stoker, J.J. (1961), Developable surfaces in the large. Comm. Pure Appl. Math. 14(3), 627--635, doi:10.1002/cpa.3160140333. A previous editor cited this reference in support of the statement "The developable surfaces which can be realized in three-dimensional space are...", implying that this is a complete list of developable surfaces. In fact the paper quotes this as an erroneous statement, then goes on to explain that there exist other developable surfaces. I don't know whether it's possible to give a complete list of 'all developable surfaces that exist in three dimensions.

For reference, here is the first paragraph of Stoker's paper:

In various textbooks on differential geometry one finds the statement

that the developable surfaces in three-dimensional space can be classified as the plane, the cylinder, the cone, or the tangent surface of a space curve. As a consequence, it would be clear that the only developables without singularities-if they are continued far enough along generators-would be the plane and the cylinder. If one examines the usual proofs of the statement, however, one finds that a quite restrictive assumption is made, i.e. that all points of the surface are either planar points (that is points such that all coefficients of the second fundamental form are zero) or that all points are parabolic points. Obviously, developables exist which contain points of both kinds. For example, a plane triangle to which circular cylinders have been attached to the sides is a case in point; and furthermore, such a surface

could be so defined as to have derivatives of arbitrarily high order.

Jowa fan (talk) 02:27, 5 December 2012 (UTC)

I don't quite get the triangle example it seems that it would need to be a surface with a boundary or self intersection. In effect his construction is union of generalised cylinders and planes. If you allow such piecewise construction then you could end up with a large set of posible shapes. --Salix (talk): 09:06, 5 December 2012 (UTC)

I hadn't realized that at all and would have removed this reference when I added the oloid to the list, which is not mentioned in the paper. --Xario (talk) 17:03, 10 December 2012 (UTC)

Extremely unclear and probably untrue assertion

In the list of examples of developable surfaces mentioned here, the following sentence:

"The oloid is one of very few geometrical objects that develops its entire surface when rolling down a flat plane"

is totally unclear as to its meaning.

And in view of the Wikipedia article on the oloid, it appears to be based on an erroneous statement in that article.Daqu (talk) 19:07, 15 November 2015 (UTC)

Prisms

If the list of allowable transformations includes "folding", then do prisms also have developable surfaces despite their cross-section curves being non-smooth? Also what is the proper term for "generalized cylinder" as defined here (and realized in the typical bath sponge) please? re/greg/ex;{mbox|history} 14:21, 23 January 2018 (UTC)

Good question, Regregex. Apparently the proper term for such a general category of shape is a "right cylinder". For example, a cylinder with two polygonal bases is called a prism; a right cylinder with two circular bases is called a right circular cylinder. 73. --DavidCary (talk) 17:50, 10 October 2020 (UTC)

Self-contradictory definition in the first two sentences

The first two sentences of the article reads as follows:

"In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression)."

As for the second sentence: No, that is not true.

Consider an infinite cylinder C that is the cartesian product of a circle and a straight line. E.g.,

C = S¹ × ℝ.

It is easy to verify that C cannot be "flattened onto a plane without distortion.

The same is true for infinitely many other surfaces with zero Gaussian curvature. For example, one nappe of a cone. Or the flat Möbius band. 2601:200:C082:2EA0:2806:FC4:E551:33CA (talk) 16:52, 10 February 2023 (UTC)