## Line Congruences (Constant Curvature II)

That revolving a simple, mechanically generated curve, the tractrix, about an axis generates the pseudosphere, a surface of constant negative curvature, seems like one of these unavoidable accidents.

The tractrix has the feature that the endpoints of its unit tangent vectors lie on a line. Thus  dragging one endpoint of a rod of length 1 will have the other endpoint trace a tractrix. More generally, whenever you have a 1-parameter family of lines in the plane, they are typically tangent to a single special curve, the caustic of the line family. Below is the caustic of the normal lines to a parabola.

In space, things get tricky. A 2-dimensional family of lines in space is called a line congruence. They also have caustics or focal sets, i.e. surfaces that are tangent to all the lines, but finding them involves a quadratic equation, so we can expect two of them. Below are the two focal sets for a hyperbolic paraboloid. It is pretty clear that line congruences are hard to visualize.

Note that some of the lines are tangent to the focal sets at some point but intersect it transversally at other points.

By rotating the lines generating the tractix, we obtain a  line congruence whose two focal sets are the pseudosphere and its rotational axis. Note that the segments of the line congruence between the two focal sets have length 1. More generally, a line congruence is called pseudospherical if the segments between the focal sets have constant length, and the surface normals at corresponding points make a constant angle. Remarkably, the focal surfaces of a pseudospherical line congruence are pseudospherical, i.e. have constant negative curvature. Even better, one can start with any pseudospherical surface, pick a point and a tangent vector at that point, and extend this vector to a pseudospherical line congruence whose first focal set is the surface one starts with. This provides a recipe to produce (essentially algebraically) new pseudospherical surfaces.

The bathtub up above is Theodor Kuehn’s pseudospherical surface from 1884. It can be obtained by a line congruence from the standard pseudosphere. Below you see a portion of the pseudosphere with asymptotic lines, and hiding behind, the corresponding portion of Kuehn’s surface.

The last image shows just the lines of the line congruence that has these two surfaces as focal sets.

Did I say this was hard to visualize?

## Revolution (Constant Curvature I)

One of the standard elementary surfaces is the Pseudosphere, a surface of revolution of constant negative curvature.

It can be parametrized using elementary function, and the profile curve is the so-called tractrix. Another elementary surface of constant negative curvature is Dini’s surface, where the tractrix is used to produce a helicoidal surface.

From here on, things get tricky. Other such surfaces of revolution require elliptic integrals. Here is the entire zoo (more or less):

Common to all examples is that they necessarily produce singularities. More precisely, there is no complete surface of constant negative curvature in Euclidean space. This is a famous theorem of Hilbert. At the core of the proofs I know is the behavior of the asymptotic lines

Above is the pseudosphere with one family of these asymptotic lines, drawn as ribbons. At the equator, they become horizontal. As the second family is the mirror image of this family, at the equator their tangent vectors become linearly dependent. This shows that while the asymptotic curves exist in the northern and southern hemipseudospheres, the surface itself is singular at the equator, because, alas, on negatively curved surfaces the asymptotic directions are linearly dependent. For the general surfaces of revolution, the asymptotic lines touch both singular latitudes. The image above looks odd because our brain wants to believe that curves on a surface meet at right angles. They don’t.

One of the key features of the asymptotic lines is that they form a Chebyshev net: Opposite edges of the net quadrilaterals have the same length. Thus you can stretch a loosely knitted square mesh over this surface to keep it warm. The standard proof of Hilbert’s theorem continues to show that any net parallelogram has area bounded above by some constant. However, a simply connected complete surface of constant negative curvature has necessarily infinite area, which leads to a contradiction. This was one of the earliest global results in differential geometry.