Differential Geometry - Computer Graphics and Geometric Modeling - page 649

(1) S is developable.
(2) S is an envelope of a one-parameter family of planes, that is, the points of S
satisfy an equation of the form (9.90).
(3) S is a ruled surface described by an equation of the form (9.85) for which
(q¢¥a)•a¢ = 0. (If a¢ π 0, then the last condition is equivalent to saying that
the distribution parameter vanishes.)
(4) The Gauss curvature of S is zero.
(5) S can be isometrically imbedded in the plane.

Proof. See [Laug65], [Lips69], [Spiv75], or [Stok69]. Note that Theorem 9.10.24 and
(4) implies (5).

Theorem 9.15.5 explains the terminology “developable” because historically these
surfaces are the ones one said could be “developed” or “rolled out” on (that is, iso-
metrically imbedded in) the plane.

9.15.6. Theorem. A necessary and sufficient condition that a curve on a surface is
a line of curvature is that the surface normals along this curve form a developable
surface.

Proof. This is a theorem of Monge. See [Will59].

9.16 The Cartan Approach: Moving Frames

Our development of differential geometry has so far basically followed the classical
approach based on local coordinates and parameterizations. In this section we would
like to sketch what a modern development would look like and discuss Cartan's theory
of moving frames.
Looking back at curves in R 3 , it turns out that a key element behind the Serret-
Frenet formulas, which is more fundamental than the actual formulas that one gets,
is the realization that relating the derivative of the Frenet frame (which measures how
it changes) to the Frenet frame uncovers a lot of geometric invariants. In the case of
surfaces, its geometry was captured by the way that the basis (F u ,F v , n ) changed for
a parameterization F, where n was the normal vector F u ¥F v normalized to unit
length. This suggests it might be useful to look at frame fields (defined below) and
how they change in general. To keep the discussion as simple as possible we shall
restrict our definitions to orthonormal bases right from the start. There is no loss in
generality because the Gram-Schmidt algorithm would let us replace arbitrary bases
with orthonormal ones anyway. Furthermore, one would move to those eventually in
any case because one needs the normalization inherent in orthonormal bases if one
wants to get geometric invariants. (Every basis of a vector space is an orthonormal
basis with respect to some inner product on that vector space and we are interested
in geometric properties with respect to a given metric.) We start with definitions for
Euclidean space.
We shall be talking a lot about vector fields in this section. Although we mostly
deal with curves and surfaces in R 3 , we again want to do things in such a way so that