Steiner Formula for Cassini Oval

Requires a Wolfram Notebook System

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

A Cassini oval is the locus of points such that , where and . If the foci and , then

[more]

, .

For the normal vector at a point on the oval,

,

where is the unit vector in the direction of .

So the normal is the diagonal of the parallelogram obtained by prolonging the vector by the magnitude of and by the magnitude of . The tangent is the line through that is perpendicular to the normal.

Let be the intersection of the perpendicular to at and the tangent. Let be the intersection of the perpendicular to at and the tangent.

Apply the law of sines to the triangle to get Steiner's formula (1835)

.

This can be applied to show that is the midpoint of the segment : the angles at and at are and , respectively, as angles with orthogonal legs. So .

[less]

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


Details

Reference

[1] A. Ostermann and G. Wanner, Geometry by Its History, New York: Springer, 2012.


Snapshots



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