Cartesian Oval

Cartesian equation:
((1 - m²)(x² + y²) + 2m²cx + a² - m²c²)² = 4a²(x² + y²)

This curve C consists of two ovals so it should really be called Cartesian Ovals. It is the locus of a point P whose distances s and t from two fixed points S and T satisfy s + mt = a. When c is the distance between S and T then the curve can be expressed in the form given above.

The curves were first studied by Descartes in 1637 and are sometimes called the 'Ovals of Descartes'.

The curve was also studied by Newton in his classification of cubic curves.

The Cartesian Oval has bipolar equation r + mr' =a.

If m = plusminus 1 then the Cartesian Oval C is a central conic while if m = a/c then the curve is a Limacon of Pascal (Étienne Pascal). In this case the inside oval touches the outside one.

Cartesian Ovals are anallagmatic curves.

Main index	Famous curves index
Previous curve	Next curve
Biographical Index	Timelines
History Topics Index	Birthplace Maps
Mathematicians of the day	Anniversaries for the year
Societies, honours, etc	Search Form

JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Cartesian.html