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.Tractrix 01

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. Paracaustic 01

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.

Kuen congruence2

Did I say this was hard to visualize?



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