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^2-4*x*z^3-4*y^3*w+6*x*y*z*w-x^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.

Mathematics 535: Introduction to Algebraic Geometry, Fall 2002

References and guide to the literature

Weekly assignments

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Week 4

Week 3

Week 2

Week 1