Cartesian sheet

Cartesian sheet for ${\displaystyle a=1}$

The Cartesian sheet (or Cartesian sheet , folium cartesii ) is a third-order planar algebraic curve named after the French mathematician and philosopher René Descartes .

definition

Let be a real number, then the Cartesian leaf in Cartesian coordinates is defined by the equation ${\displaystyle a\in \mathbb {R} }$

${\displaystyle x^{3}+y^{3}-3axy=0\,.}$

Other cartesian leaf equations

In parametric representation , the Cartesian sheet can be represented by the equations

${\displaystyle {\begin{aligned}x={\frac {3at}{1+t^{3}}}\\y={\frac {3at^{2}}{1+t^{3}}}\end{aligned}}}$

described, where is. ${\displaystyle a\in \mathbb {R} }$

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

${\displaystyle r={\frac {3a\cos \varphi \sin \varphi }{(\cos \varphi )^{3}+(\sin \varphi )^{3}}}}$

described.

Properties of the Cartesian sheet

In the following it is assumed that the coordinate axes are as shown in the sketch.

• The Cartesian sheet is axially symmetrical with respect to the bisector of the 1st and 3rd quadrant (equation y = x). Exactly two points of the curve lie on the symmetry axis, namely the origin and the vertex with the coordinates . ${\displaystyle \left({\tfrac {3}{2}}a\left|{\tfrac {3}{2}}a\right.\right)}$
• The origin of the coordinate system is the colon of the curve. The x and y axes are the respective tangents at the origin.
• The straight line with the equation (dashed blue in the sketch) is the asymptote of the curve. ${\displaystyle x+y+a=0}$
• For both branches of the curve the radius of curvature is at the origin . ${\displaystyle {\tfrac {3}{2}}a}$
• The loop of the Cartesian leaf encloses an area with the content . ${\displaystyle {\tfrac {3}{2}}a^{2}}$
• The area that is bounded by the curve and the asymptote and that extends to infinity has the same area . ${\displaystyle {\tfrac {3}{2}}a^{2}}$

Web links

• Eric W. Weisstein : Folium of Descartes . In: MathWorld (English).
• John J. O'Connor, Edmund F. Robertson :  Folium of Descartes. In: MacTutor History of Mathematics archive .