For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse aligned with the principal directions.
For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbola will be formed on either side of the tangent plane. These hyperbola share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions.