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The Hessian

The hessian of a hypersurface is the determinant of the matrix of its second partial derivatives. The hessian of a plane curve intersects the curve exactly in those points at which a tangent to the curve has intersection multiplicity greater than 2 with the curve, e.g. at flex or singular points. An ordinary flex point is a point in which the curvature of the curve changes its sign. Similarly, the hessian of a surface in 3-space intersects the surface in a curve which separates regions of positive curvature from regions of negative curvature. We show a family of plane cubics in the spherical view of the projective plane and a family of cubic surfaces in affine 3-space. Both families contain a singular example. An advantage of the spherical view of the projective cubic curve of the form y^2=f(x) is that we can see all three real flex points of the plane cubic. In the affine view, one of them is at infinity. This film was made by Oliver Labs using surfex.