In this article, we shall describe elliptic, hyperbolic, parabolic points of a surface. We choose a torus as the surface for this purpose because it is a good example where these points occur in various parts of the surface.

Consider an elliptical torus:

The parametric form of an elliptical torus is given by

torus[a,b,c](u,v)=((a + b cos v) cos u, (a + b cos v) sin u, c sin v),

where u,v ∈ [0, 2π).

Here is a graph of a torus with a=2, b=1, c=1:

By fixing the *v*-parameter, the following graph shows several *u*-coordinate curves on the torus[2,1,1]:

The Gaussian curvature of the torus can be computed using the equation

,

where E, F, G are the coefficients of the first fundamental form relative to the torus, and e, f, g are the coefficients of the second fundamental form.

Writing

** x**(u,v)≡torus[a,b,c](u,v), and denote subscript as partial derivative,

we have

, , .

and

where U is the unit normal vector at points on the torus.

It is interesting to note that the Gaussian curvature for the torus depends only on v

As for our example, consider a torus with *a**=**2*, *b**=**1*, *c**=**1*, where u,v ∈ [0,2 π). The Gaussian curvature for this torus becomes,

.

Plotting the Gaussian curvature with respect to *v*, we obtain

Points on the surface is called elliptic if the Gaussian curvature is positive, that is, for our case

This occurs when

and .

So, the elliptic points lie on these curves

Points on the surface is called hyperbolic if the Gaussian curvature is negative, that is, for our case

This occurs when

The following graph shows hyperbolic points that lie on some of the curves.

The shape operator of the **torus**[2,1,1] can be computed and found to be nonzero

Points on the surface is called parabolic if the Gaussian curvature vanishes and the **shape operator** is nonzero.

This occurs when

and .

There are only two curves that lie on the **torus**[2,1,1] which are parabolic as is shown in the following graph:

Finally the figure below shows curves on the **torus**[2,1,1] where points are elliptic (blue curves), hyperbolic (green curves), and parabolic (red curves).

REFERENCES:

Gray, A., Abbena, E., Salamon, S. Modern Differential Geometry of Curves and Surfaces with Mathematica, 3nd ed. Boca Raton, FL: CRC Press, 2006.

Tan, S., Vector Calculus Using Mathematica, 1st ed. Lulu, 2018.