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The Cartesian Ovals

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Notes

  1. Descartes confusingly employs both letters and numbers to label points. Thus, for example, the line segment from the point 6 to the point R in the diagram will be denoted by 6R.

  2. We consider only circles or disks centered at the point 1, since the results for arbitrary (nonzero) centers are simply scaled/rotated versions of the resulting Minkowski products.

  3. The envelope of a variable circle whose center lies on the circumference of another circle and whose radius is proportional to the distance of its center from a fixed point is a pair of ovals of Descartes.

  4. The extended complex plane is the set of all finite complex values augmented by a single point \(\infty \) at infinity.

  5. For brevity, we consider problems that are rotationally symmetric about a certain axis; the wavefront surfaces are then completely characterized by planar sections.

  6. The name “caustic” is due to Ehrenfried Walther von Tschirnhaus, a contemporary of Leibniz and Newton, who investigated the use of mirrors to focus sunlight: the Latin and Greek roots of “caustic” mean capable of burning.

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  1. Department of Mechanical and Aerospace Engineering, University of California, Davis, CA, 95616, USA

    Rida T. Farouki

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Farouki, R.T. The Cartesian Ovals. Math Intelligencer 44, 343–353 (2022). https://doi.org/10.1007/s00283-021-10149-8

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