題目如下
A natural generalization of both surface of revolution and helicoids is
obtained as follows. Let a regular plane curve C, which does not meet an axis E
in the plane, be displaced in a rigid screw motion about E, that is so that
each point of C describes a helix(or circle) with E a axis. The set S generated
by the displacement of C is called a generalized helicoid with axis E and
generator C. If the screw motion is a pure rotation about E, S is a surface of
revolution; if C is a straight line perpendicular to E, S is (a piece of)the
standard helicoid choose the coordinate axes so that E is the z axis and C
lies in the yz plane.
Prove that
(a).If(f(s),g(s)) is a parametrization of C by are length s, a0,
then x:U->S, where U={(s,u)<-(R^2); a
x(s,u)=(f(s)cosu,f(s)sinu,g(s) cu),
c=const., is a parametrization of S.
Conclude that S is a regular surface.
(b).The coordinate lines of the above parametrization are orthogonal
(i.e.,F=0)if and only if x(U) is either a surface of revolution or (a piece of)
the standard helicoid.
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