5.8 Osculating Paraboloid

When considering the structure of a surface at a point, it is often helpful to approximate the surface using another surface whose behavior is more easily visualized. A spherical surface would be insufficient for this purpose because it has the same normal curvature in all directions. We can, however, use the osculating paraboloid introduced in (5.30). Let us reconsider this paraboloid from the point of view of local approximation.

Let O be a fixed point of a surface, and assume the surface is sufficiently smooth at O. At O we determine the osculating plane and introduce a Cartesian frame ( i ₁, i ₂, n), where ( i ₁, i ₂) is a Cartesian frame with origin O on the osculating plane and n is normal to both the surface and the osculating plane. For a smooth surface Cartesian coordinates ( x, y) can play the role of surface coordinates since they uniquely define any point of the surface at O. The surface in the vicinity of O can be described by the equation z=z(x, y). We suppose that z(x, y) is twice continuously differentiate near O. In the neighborhood of (0, 0) we can use the Taylor expansion of z(x, y), which is

where the indices x, y indicate that we take partial derivatives with respect to x, y, respectively (and in this section, evaluated...