Polar Curves

The polar curve PC,p from a point p to a plane curve C of degree d is a plane curve of degree d-1 which intersects the original curve C exactly in those points ti for which the line through p and ti is a tangent to C through p.

In our example, you see a one parameter family of projective cubic plane curves (black) in the spherical view, together with its polar (white) from the green point, and the (red) tangents from the point to the black curve.

When a tangent at a flex point of the (black) curve passes through the green point, or when an ordinary double point develops, one can see that two of the intersection points of the polar (white) with the curve (black) coincide. The same thus also holds for two red tangents from the green point to the black curve. Moreover, when a cusp singularity develops, four of the intersection points and thus four red tangents coincide.

The point in which the two white great circles intersect, i.e. the singular point of the polar, is a flex point of the (black) curve. The (red) tangent at that point passes through the green point throughout the whole animation.

This film was made by Oliver Labs using Singular and surfex.