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

5.1. Ruled surfaces 187
Figure 5.2: The surface of binormals of the Viviani’s temple
The relation 5.1.1 provides a parameterization of the ruled surface,
r : I × R → S.
This parameterization is not, usually, global, as the parameterization of the directrix is
not global.
With respect to this parameterization, the rulings will be coordinate lines (u = const),
and the directrix is, equally, a coordinate line (v = 0). Generally, the coordinate lines
v = const have the property that they are “parallel” to the directrix, in the sense that all
the points from such a coordinate curve lie at the same distance (equal to |v|) from the
directrix, when we measure the distance along the ruling passing through each point.
The tangent plane and the first fundamental form of a ruled surface
To compute the coefficients of the first fundamental form of a ruled surface we need,
first of all, the partial derivatives of the radius vector of a point of the surface. We have,
obviously,
r ′ u = ρ ′ + bb ′ ; r ′ v = b ′ u. (5.1.2)