Asymptotic Curves

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:

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.

X 10aX 10aX 10a

 

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.

X 10bX 10bX 10b

 

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

X 10c X 10c X 10c

 

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

 X 10d X 10c X 10d

 

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

X 10fX 10fX 10f

 

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

X 10gX 10gX 10g

 

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

X 10hX 10hX 10h

 

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.

X 10iX 10iX 10i

 

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 G√∂ttingen's undamaged model

X 10kX 10kX 10k

 

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. 

X 10lX 10lX 10l

 

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 $1$.

X 10mX 10mX 10m