Folium of Descartes

The Cartesian sheet (or Cartesian leaf, folium cartesii ) is a plane algebraic curve of 3rd order, which is named after the French mathematician and philosopher René Descartes.

Definition

Let be a real number, then the Cartesian leaf is in Cartesian coordinates defined by the equation

Other equations of the Cartesian leaf

In parametric form, the Cartesian leaf by the equations

Will describe being.

In polar coordinates, the Cartesian sheet is given by the equation

Described.

Properties of the Cartesian leaf

In the following, each case that the coordinate axes are as shown in the sketch.

• The Cartesian sheet is axisymmetric with respect to the bisector of the first and third quadrants ( y = x). Exactly two points on the curve lying on the axis of symmetry, namely the origin and with the coordinates of the vertices.

• The origin of the coordinate system is the double of the curve. The x-and the y-axis, the respective tangents at the origin.

• The straight line with the equation ( in the sketch blue dashed line) is the asymptote of the curve.

• Both branches of the curve radius of curvature is at the origin.

• The loop of the Cartesian sheet includes an area with the content.

• The area which is bounded by the curve and the asymptote and extending to infinity, has the same surface area.