We then tried another experiment that produces curves from straight lines. In this activity each student puts a dot on a large sheet of paper; their dot must “chase” another student’s dot by moving straight towards it by fixed distance each time. The resulting lines start out straight but, since every dot is moving, soon begin to curve and spiral around each other.

This gives us a different and complementary idea of the tangent space as a description of motion: specifically, the trajectory a point is moving on at an instant. Incidentally, you can see some striking computer visualisations of pursuit curves in this USAF video designed to train pilots in dogfight tactics:

To clarify our ideas we turned to simple examples: a straight line, whose tangent spaces are “all the same”, and a circle, whose tangent spaces “turn at a constant rate”. We decided we could use circles of different sizes to measure the curvature of our more irregular lines, and reached the notion of osculating circle. This brings together the geometric idea of curvature with the more dynamic, physical one; we imagined driving along a mountain road and considered the turning circle of the car as it navigated the bends. On the left are one student’s notes from class illustrating this idea.

This was what we needed from this class; next week we go up a dimension and look at the Gaussian curvature of 2-manifolds. This will lead us to hyperbolic geometry, the last of the three geometries we meet on this course. In the final session we’ll return to 1-manifolds with a session on knots, which happen to have both curvature and torsion, but there our interest will be purely topological.