Blaga P. Lectures on the differential geometry of - tiera.ru

190 Chapter 5. Special classes of surfaces
Proposition 5.1.2. The total curvature of a ruled surface given by the local parameterization
(5.1.1) is given by
Kt = DD′′ − D ′2
EG − F
2 = −
(ρ ′ , b, b ′ ) 2
The curvature is always negative and it vanishes if and only if
5.1.3 Envelope of a family of surfaces
�(ρ ′ + vb ′ . (5.1.7)
) × b�2 � ρ ′ , b, b ′ � = 0. (5.1.8)
In this section, a surface will always be a parameterized surface. The theory of envelopes
of surfaces is analogue to the theory of envelopes of plane curves, therefore there will
be no proofs, as the proofs are completely analogue to the ones we provided for plane
curves.
Definition 5.1.1. Let
r = r(u, v, λ) (5.1.9)
be a family of surfaces. The envelope of the family (if there is any) is a surface which is
tangent to all the surfaces from the family. The contact between the envelope and each
surface is made along a curve which is called a characteristic. Thus, the envelope is the
geometrical locus of the chracteristics of the family of surfaces.
Proposition 5.1.3. The points of the envelope of the family (5.1.9) verify, beside this
equation, the equation
(r ′ u, r ′ v, r ′ λ ) = 0. (5.1.10)
Of course, as in the case of plane curves, these equations are also verified by the
singular points of the surfaces and to get the envelope, we have to eliminate them first.
If the family of surfaces is given implicitly, i.e. through the equation
F(x, y, z, λ) = 0, (5.1.11)
then to get the envelope, we have to add to this equation the equation
F ′ λ (x, y, z, λ) = 0, (5.1.12)
as we did in the case of plane curves. Something that is specific to the theory of envelopes
of surfaces is the so-called regression edge. Let us assume that we have a family of