Osculating Paraboloids

Synonyms

Osculating quadric; Second order approximation

Related Concepts

Curvature; Differential Invariants

Definition

Osculating paraboloids commonly occur in second-order approximations where the first order is irrelevant or can be transformed away.

Background

Despite the common occurrence of osculating paraboloids in many settings, there appears to be no literature dedicated to their shape space. Although the standard taxonomy of quadrics in Euclidean space is familiar to most, what is missing is the geometrical structure of the manifold of all osculating paraboloids, inclusive a metric. Here the two-parameter case (important in many applications) is discussed in a little detail.

Theory

“Osculating paraboloids” are second-order Monge patches:

$$\begin{array}{rcl} & x \mathbf{e}_x+y \mathbf{e}_y+z(x,y)\mathbf{e}_z =x \mathbf{e}_x \nonumber\\ & +y \mathbf{e}_y+\frac{1}{2}(a_{20}x^2+2a_{11}x y+a_{02}y^2)\mathbf{e}_z, \end{array}$$

(1)

that occur in many contexts. Apparently they...