The Kappa Curve as a Locus

The kappa curve looks vaguely like a curly kappa, , and was studied by Newton, Bernoulli, and Gutschoven. Sometimes called Gutschoven's curve, its double cusp form can be represented as .
Let be the point (magenta). Let be the intersection point (blue) of the horizontal line with a line rotating about the origin . The kappa curve is the locus of points (red) on the rotating line such that distances (green segments).


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[1] T. Heard, D. Martin, and B. Murphy, A2 Further Pure Mathematics, 3rd ed., London: Hodder Education, 2005 p. 209, question 13.


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