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

194 Chapter 5. Special classes of surfaces
Figure 5.4: The tangent developable of Viviani’s temple
We shall assume, without restricting the generality, that the normal vector N is a versor.
By differentiating the equation (5.1.15) with respect to λ, we obtain
N ′ (λ) · r + D ′ (λ) = 0. (5.1.16)
According to the theory of envelopes of the families of surfaces, the equations (5.1.15)
and (5.1.16) determine the envelope of the family of planes, if such an envelope exists 2 .
The characteristic curves of the family of planes are straight lines, obtained as intersections
between the planes of equations (5.1.15) and (5.1.16), respectively. Clearly,
these characteristics exist only if the planes of the family are not parallel, which we
admit to be true in our case.
The points of the envelope have to verify, also, the equation obtained from (5.1.16),
by differentiating once more with respect to λ, i.e.
We have, thus, the following system of equations:
⎧
N(λ) · r + D(λ) = 0
⎪⎨
N
⎪⎩
′ (λ) · r + D ′ (λ) = 0
N ′′ (λ) · r + D ′′ (λ) = 0.
N ′′ (λ) · r + D ′′ (λ) = 0. (5.1.17)
(5.1.18)
2 As a matter of fact, these equations describe the discriminant set, including, also, the singular points
of the surfaces from the family. However, in this particular case, the surfaces are planes and they have no
singular points whatsoever.