4. Constructing a Point on a Cassini Oval

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows another construction of Cassini's oval. Start with the hyperbola with equation of eccentricity , . Select any point on . Let be the opposite point of and a point on different from and . The tangents on at and are parallel and meet the tangent at at points and , respectively. Then .

[more]

Draw a circle with center and radius and a circle with center and radius ; suppose these meet in points and . But then . So is a point on a Cassini oval with foci and . The same is true for the point . It can be shown that the foci and are also on the oval.

[less]

Contributed by: Marko Razpet and Izidor Hafner (July 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

In Conics, Book III, theorem 42, Apollonius has shown that for and for the ellipse.

References

[1] T. Heath, A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus, New York: Dover Publications, 1981, p. 155.

[2]. A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012, pp. 76–78.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send