Inverted Cassini Ovals

Michael Williams and Dr. Ivona Grzegorczyk


The inscribed angle theorem tells us that an angle inscribed in a circle has one half the measure of the central angle of the circle that subtends the same arc. Another way to think of this is that all the points which view a segment at a constant angle will lie on the arc of a circle or its reflection. This statement has been generalized to account for two segments and results in a  curve known as the Apollonian cubic. This research further generalizes the Apollonian cubic to a curve with arbitrary angle. The main question is, what are the points that view two segments with oriented angles having constant sum? Using techniques from complex analysis and ideas from inversive geometry we find, in general, that the points lie on the circle inversion of a Cassini Oval. Cassini Ovals are important curves in mathematics with applications in astronomy, nuclear physics, acoustics, and other sciences.


  1. Congratulations Michael!

    Nice visuals! Can you give some additional applications? The question itself is very interesting, thanks for sharing your solution.

    • Dr Flores,

      Thanks for your comment. As for your question, I know of some applications, but they are a bit more general. There is a question that asks whether or not a shape can be determined from its isoptic. The answer is generally no (there is an ellipse and a circle which have the same (pi/2)-isoptic). But, to my knowledge, this is unknown for two objects. So in a sense this work can be applied to some inverse / reconstruction problems based on viewing angles.

      Other than that, I imagine there might be some application in object detection. For instance if you have an object flying above two sensors, you could calculate the angles subtended by the sensors at the object to learn what cassini oval it is on and therefore its position. In a way, we can create a coordinate system based on the two sensors.

      Hope this answers your question. I’d gladly talk to you more about it if you want.

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