8.4: Elliptic, Parabolic, and Hyperbolic PointsDefined in terms of the Second Fundamental FormA direction in the tangent plane at a point for which the normal curvature is zero is called an asymptotic direction. In terms of the second fundamental form, such a direction satisfies the condition
where λ = v'(t)/u'(t).
If we have a curve on the surface defined by X(t), with a non-zero
tangent vector, this condition can be written as
If at some point P of a given surface det(Lij) > 0 we call P an elliptic point. Clearly at such a point P there can exist no real asymptotic directions. If det(Lij) = 0 at P , P is called a parabolic point of the surface. Parabolic points can only have one asymtotic direction. For example, all points on a cylinder are parabolic and only have one asymptotic direction, namely the straight line. If det(Lij) <0 at P , P is called a hyperbolic point and P will have two asymptotic directions. Defined in terms of Gaussian Curvature While many differential geometry books use the definitions provided above for elliptic, parabolic and hyperbolic points of a surface in three-space, others define elliptic, parabolic and hyperbolic points of a surface to be points where the Gaussian curvature is strictly positive, zero and strictly negative, respectively. The connection between the two definitions can be explained using the Weingarten map. The coefficients of the second fundamental form are related to those of the Weingarten map by the relation (summing over repeated indices)
Asymptotic Directions Demonstration
Depending on whether we take the plus sign or the minus sign in the above equation for the asympotic directions, we can have different sets of vectors. In this demo, we can show one set, or the other, or both. These display options are controlled by the Show Vec Field 1 and Show Vec Field 2 checkboxes respectively. We can move around a point in the domain, and adjust the tangent directions of two curves on the surface. The curves are shown on the surface, along with each of their normal curvature vectors κNN. Line up the tangent directions with the vector field to see that the normal curvature is zero in the asymptotic directions. (NOTE: does not work for horizontal and vertical directions). Exercises
2. On the torus, what happens to the asymptotic curves between regions of positive and negative Gaussian curvature? Why? |