Many of our surfaces of revolution exhibit "asymptotic curves" (also referred to as "asymptotic lines"). These are defined in an analogous manner to geodesics. A geodesic has constant zero geodesic curvature, while an asymptotic curve has constant zero normal curvature (this is a somewhat more complicated notion of curvature - see MathWorld for an explanation).

Asymptotic curves are only defined at points where the Gaussian curvature is negative. So, on surfaces of positive curvature, there are no asymptotic curves, but there are two asymptotic curves through every point on a surface of negative curvature. The asymptotic directions of these two curves are each bisected (i.e. the angles of intersection are split into two equal parts) by the principal directions. This can be seen on model X 10h:

As with lines of curvature and principal directions, the asymptotic lines determine "asymptotic directions" (or equivalently, are themselves determined by the asymptotic directions). Recall our notion, $\kappa_{E}$, of curvature from when we were defining principal curvature in Differential Geometry. Asymptotic curves can be thought of as a kind of "boundary" between positive and negative values of the curvature $\kappa_{E}$ as we rotate the plane $E$. At a point with negative Gaussian curvature, rotating the plane from one principal direction to another changes the sign of the curvature $\kappa_{E}$. Hence (by the intermediate value theorem) the curvature must at some point satisfy $\kappa_{E} = 0$. The two directions defining planes for which $\kappa_{E} = 0$ are our asymptotic directions. Note that if the Gaussian curvature is non-negative then this notion is not well-defined, since the planes all have $\kappa_{E} \geq 0$.

Asymptotic directions can also be realised as the asymptotes of the Dupin indicatrix at points with negative curvature. The Dupin indicatrix is defined by the intersection of a surface with an arbitrarily close plane parallel to the tangent plane. It is a hyperbola for points of negative curvature. See here for more on asymptotic lines, with nice visuals.

All of the following models exhibit lines of curvature and asymptotic curves. The asymptotic curves are distinguishable as, informally speaking, those which spiral around the surface, while the lines of curvature are contained either in a vertical or horizontal plane.

**Model X 10a **is generated by rotating the quadratic $z^2 = x$ about the $z$ axis. The minimum point of the quadratic curve meets the axis of rotation in a singularity - this should be in the gap at the centre of the model, but has deteriorated since it was so thin.

**Model X 10b** is obtained by rotating the cubical parabola $z^3 = 27x$ about the axis tangent to its point of inflection (i.e. about the $z$-axis). The point of inflection meets the axis of rotation in a singularity - again this should be in the gap at the centre of the model, but has deteriorated since it was so thin.

**Model X 10c**** **is obtained by rotating $z= 8/x^2$ about the $z$-axis.

**Model X 10d **is generated by rotating the rectangular hyperbola $z=6/x$ about the $z$-axis. Again, our copy is rather decrepit.

**Model X 10f **is obtained by rotating the semicubical parabola $z^3 = 25x^2$ about the $z$-axis. Note this is displayed upside-down.

**Model X 10g **is obtained by rotating the parabola $z^2 = a^2 (x-a)$ about the $z$-axis.

**Model X 10h** is obtained by rotating the cubic parabola $z^3 = a^3 (x-a)$ about the $z$-axis.

**Model X 10i** is obtained by rotating the semicubical parabola $z^3 = a^3 (x-a)^2$ about the $z$-axis, for negative $a$. This model shows the 'top' of the surface, and continues in the downwards vertical direction to infinity. The 'rim' (now badly damaged) shows the path traced by the cusp of the semicubical parabola as it rotates about the $z$-axis.

**Model X 10k **is obtained by rotating the parabola $z=a(x-a)^2$ about the $z$-axis. The model should feature another small piece sitting on the top, as seen here with Göttingen's undamaged model.

**Model X 10l **is defined by the more complicated equation $z=(\sqrt{c^2 -x^2} - c^2 \arccos(x/c))/2c$. The "projection" of the asymptotic curves onto the plane $z=0$ is particularly interesting - they are all circles. This is exactly analogous to taking a photo from above - see the third image.

**Model X 10m **is generated by rotating $z= \cos x$ about the $z$-axis. Only one and a half periods of cos are shown on the model, with the circles displaying the parts of the surface where $z$ takes value $-1, 0,\ $or $1$.