To this end, we examine again the surface of revolution
u vfucos v fusin v gu
obtained by rotating the unitspeed curve u
fugu in the xzplane around the zaxis. We found in Example that its Gaussian curvature is
f f
K
Suppose first that K everywhere. Then, Eq gives f so fu au b for some constants a and bSince f g we get g a
so we must have a and hence gu aucwhere c is another
constant. By applying a translation along the zaxis we can assume that c and by applying a rotation by about the xaxis, if necessary, we can assume that the sign is This gives the ruled surface
u vb cos v b sin v ua cos v a sin v a
If a this is a circular cylinder; if a it is the xyplane; and if a it is a circular cone to see this, put u au b Now suppose that Ksay K Rwhere R is a constant. Then,
Eq becomes f R f
which has the general solution fu a cos
u R b
where a and b are constants. We can assume that b by performing a reparametrization u u Rbv v Then, up to a change of sign and adding a constant,
gu
a R sin u
R gu R sin u R du The integral in the formula for gu can be evaluated in terms of elementary
functions only when a or RThe case a does not give a surface, and if a R then fu Rcos u
R and we have a sphere of radius
R the case a R can be reduced to this by rotating the surface by around the zaxis Suppose finally that K We can restrict ourselves to the case K
as the general case can be obtained from this by applying a dilation of Rsee Exercise In view of the preceding case, we can think of a surface with K as a sphere of imaginary radius
or a pseudosphere
