In case r = 0 we obtain the unit 2-sphere
[S.sub.2], but for r > 0 the surface S is no longer differentiable at points whose u-coordinate is 0.
For t = 0, we obtain a fixed-point configuration, q(0), specific to [S.sup.3], in the sense that there is no 2-sphere
that contains it:
where [theta] is a constant (angle) different from 0, [xi] = [xi](u) = cot [theta] log u and f is a unit speed curve on the Euclidean 2-sphere
This is the line-element of a 2-sphere
with a radius of curvature of [square root of 2b], i.e.
The subject of many areas of investigation, such as meteorology or crystallography, is the reconstruction of a continuous signal on the 2-sphere
from scattered data.
To see that W = [[Sigma].sup.3] - Int N(L) is irreducible, suppose that F is a smoothly embedded 2-sphere
in the interior of W and let [D.sub.1] and [D.sub.2] be the closures of the two components of [[Sigma].sup.3] - F in [[Sigma].sup.3].
It is well known that [[??].sub.0] of the coordinate ring A = R[X, Y, Z]/([X.sup.2] + [Y.sup.2] + [Z.sup.2] - 1) of the algebraic 2-sphere
[S.sup.2](R) over a field R is isomorphic to the integers Z, whenever R is of characteristic not two, and contains i, the squared root of -1, see [6, Corollary 10.8] and [7, [section] 7].