$\begingroup$

I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?

$\endgroup$ 2

3 Answers

$\begingroup$

It's called "the saddle" :). Substitute $x=(u+v)$ and $y=(u−v)$ to get $z=(u+v)(u−v)=u^2−v^2$, which is a more conventional parametrization of the surface, while the surface itself is unchanged.

$\endgroup$ 11 $\begingroup$

It is simply a hyperbolic paraboloid, equivalents to the surface $z=x^2-y^2$. Its cross-sections are parabolas and hyperbolas.

$\endgroup$ $\begingroup$

Saddle point is an attribute/ character of all surface points. If you hold a 3D model of it in your hands, its nomenclature is invariant by the direction of your view :) but lines of projection can have a separate name.

EDIT 1:

Sorry did not follow OP properly at that time, (now deleted phrase hyperboloid of 1 sheet). All points of negative Gauss curvature have Saddle points qualitatively, as against the ellipsoidal points. (synclastic/anticlastic etc.)

The real parts of $ ( x+ i y)^2 = ( x^2 -y^2+ i \,2 x y )\quad \, ( z= 2 xy ;\, z= x^2 -y^2 ) $ are intrinsically same, one can be obtained from the other by rotation through$ 45^0 $ about z-axis.Are ruled surfaces. The surface is called a hyperbolic paraboloid due to cross sections as parabolas/hyperbolas. Hypar is a term used in Civil engineering application.

AFIK there is no standard saddle implying a fixed geometrical parametrization and there perhaps need not be. It is just contrasted from convex bulbous surface geometries.

$\endgroup$ 3

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy