Folium - Encyclopedia

GEOGRAPHICAL NAMES

FOLIUM, in mathematics, a curve invented and discussed by Rene Descartes. Its cartesian equation is x 3 -1-y 3 =3axy. The curve is symmetrical about the line x = y, and consists of two infinite branches asymptotic to the line x+y+a = o and a loop in the first quadrant. It may be traced by giving m various values in the equations x=3am/ ('1-1-m 3 ),' y=3am2 (1-1-m 3), since by eliminating m between these relations the equation to the curve is obtained. Hence it is unicursal (see CURVE). The area of the loop, which equals the area between the curve and its asymptote, is 3a/2.