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Published on Dec 12, 2006
At the left, you see a projective cubic plane curve (black), together with its tangents (white). At the right, you see its dual curve (red) which consists of those points in the dual projective plane which correspond to the tangent at a given point of the original curve, together with the tangent at that point (green).
At the node (the double point) of the black curve, one seees that there is not a unique tangent at that point. In the dual curve, this is reflected by the fact that the tangent at the corresponding point touches it in two points.
Similarly, once the point on the original curve in which we draw the tangent passes the flex point, the tangent changes its direction. This is reflected on the dual curve by the fact that the curve changes its direction, i.e. it has a cusp singularity.