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:

        torus

 

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

         torus

 

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

            index_1.png,

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 

       index_2.png,    index_3.png,    index_4.png.

and

        index_5.png

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

        index_6.png

 

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,

        index_7.png.

 

Plotting the Gaussian curvature with respect to v, we obtain

         curvature of torus

 

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

            index_8.png

This occurs when
        index_9.png    and    index_10.png.

So, the elliptic points lie on these curves

         elliptic points on torus

 

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

            index_11.png

This occurs when
            index_12.png

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

         hyperbolic points on torus

 

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

            index_13.png

 

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

This occurs when
        index_14.png    and    index_15.png.

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

        parabolic points on torus

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).

        elliptic, hyperbolic, and parabolic points on torus

REFERENCES:

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