Osculating paraboloid

of a surface at a point $P$

The paraboloid that reproduces the shape of the surface near this point up to variables of the second order of smallness with respect to the distance from $P$. Let $\Phi$ be a paraboloid (see Fig.) with vertex $P$ and tangent to the surface at this point, and let $h$ and $d$ be the distances of an arbitrary point $Q$ on the paraboloid to the surface and to $P$, respectively.

Figure: o070550a

Then $\Phi$ is said to osculate if $h/d^2\to0$ as $Q\to P$. This does not exclude the degeneration of the paraboloid into a parabolic cylinder or plane. At every point of a regular surface there is a unique osculating paraboloid. Osculating paraboloids can be used to classify the points on a surface (see Elliptic point; Hyperbolic point; Parabolic point; Flat point).

Comments

The osculating paraboloid at $P$ to the surface $S$ has contact of order three with $S$ at $P$, i.e. the derivatives up to and including order 2 of the difference $p(x,y)-s(x,y)$ of the functions $p(x,y)$ and $s(x,y)$ describing the paraboloid and the surface are all zero at $(x_0,y_0)$, where $P=p(x_0,y_0)=s(x_0,y_0)$.

References

[a1]	R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 138
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)