Re: What is "development of a surface" called in math lingo?

joshipura@xxxxxxxxx writes:

I am an engineer. I studied something called "development of surfaces".
I want to know
1. what it is mathematically called

"Development of surfaces" and similar terms (you might search on
"tangent developable", for example).

and
2. whether there exists any site that explains the mathematics behind
it in layman's language.

Sorry, no idea.

The idea was to cut the *** metal and then fold it in a way it
generates given surface. For example:
1. A cube is "developed" as four squares in line and two other squares
with one side common to any one of the four
2. A cylinder is "developed" as a rectangle with length equal to the
circumference and height equal to the height of the cylinder
3. A cone is "developed" as an arc with its radius equal to the
generator of a cone and arc that is a proportionate to the ratio of
base radius / generator
4. A sphere or a torus could never be "developed"
(Engineering graphics did not have more complicated surfaces :-)

Of these, only (1) is outside the standard use of the word "develop"
(and its derived forms) in mathematics, but even in case (1) I have
heard the word used (by Dennis Sullivan, c. 1971; I don't know if he
was basing his usage on prior usage, or making a generalization--this
was at the same time he was developing a theory of differential forms
on polyhedra, and other similar generalizations of "differential"
constructions to "piecewise-linear" contexts, so it's perfectly possible
that he did make it up himself; on the other hand, it was also a period
when he advocated, and acted on, the principle that modern mathematicians
could learn a lot by reading old papers, not necessarily always the
"classics", so it's also perfectly possible that he found it somewhere
in a minor work of Cauchy, say). Any (convex) polyhedral surface can
be "developed" by setting it down with one face in a plane, then
rolling it over some edge of that face until the adjacent face is
in the plane, and repeating, at each stage copying the face that
is on the plane onto a polygonal region in that plane; of course
in general this will result in overlaps, but in favorable cases
(like the cube), for a good choice of successive edges, the process
will create a (generally non-convex; but consider the tetrahedron)
polygonal region in the plane which can be "cut out" and "folded up"
to recreate the original polyhedron. The flat metric of the plane
is preserved everywhere (even along the edges) except at the vertices.
Even there you can give the polyhedron a compatible conformal structure.
"Development" in smooth cases is similar.

Lee Rudolph
.

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