The picture at left shows an affine view of a projective
surface which is smooth at all points except those on the curve
forming a ridge in the middle of the surface. This surface is the
tangent developable to the twisted cubic curve, that is the union of
all the tangent lines to the twisted cubic. The projective equation
is 3*y^2*z^24*x*z^34*y^3*w+6*x*y*z*wx^2*w^2. One can check that
the points in projective space for which all partial derivatives
vanish are precisely those on the twisted cubic curve. Several lines
are drawn
on the
surface to illustrate how it is swept out by tangent lines to the
cubic.
We have drawn the real locus in the affine space w=1, clipped to fit
inside a spherical region. Such real developable surfaces have the
property that they can be assembled out of pieces of sheet metal,
since they have zero curvature, generalizing the construction of
cylinders or cones by rolling up a sheet. In this example, two flat
sheets can be cut such that they are bent and then soldered along the
twisted cubic to yield this surface. You can see the tangents as
bands of light along the brass interior of the surface.

I last taught this course several years ago. The old course web site gives an idea of what the course covered then.