# Folium of Descartes

Last updated The folium of Descartes (green) with asymptote (blue) when a = 1.

In geometry, the folium of Descartes is an algebraic curve defined by the equation Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a higher dimensional space or defined over some more general field.

## Contents

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

It forms a loop in the first quadrant with a double point at the origin and asymptote In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.

$x+y+a=0\,$ .

It is symmetrical about $y=x$ .

The name comes from the Latin word folium which means "leaf". Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets and ultimately from the Phoenician alphabet. A leaf is an organ of a vascular plant and is the principal lateral appendage of the stem. The leaves and stem together form the shoot. Leaves are collectively referred to as foliage, as in "autumn foliage".

The curve was featured, along with a portrait of Descartes, on an Albanian stamp in 1966.

## History

The curve was first proposed by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.  Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Graphing the curve

Since the equation is degree 3 in both x and y, and does not factor, it is difficult to solve for one of the variables.

However, the equation in polar coordinates is:

$r={\frac {3a\sin \theta \cos \theta }{\sin ^{3}\theta +\cos ^{3}\theta }}.$ which can be plotted easily. By using this formula, the area of the interior of the loop is found to be $3a^{2}/2$ .

Another technique is to write y = px and solve for x and y in terms of p. This yields the rational parametric equations: 

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L.

$x={{3ap} \over {1+p^{3}}},\,y={{3ap^{2}} \over {1+p^{3}}}$ .

We can see that the parameter is related to the position on the curve as follows:

• p < -1 corresponds to x>0, y<0: the right, lower, "wing".
• -1 < p < 0 corresponds to x<0, y>0: the left, upper "wing".
• p > 0 corresponds to x>0, y>0: the loop of the curve.

Another way of plotting the function can be derived from symmetry over y = x. The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis.

This operation is equivalent to a substitution:

$x={{u+v} \over {\sqrt {2}}},\,y={{u-v} \over {\sqrt {2}}}$ and yields

$v=\pm u{\sqrt {\frac {3a{\sqrt {2}}-2u}{6u+3a{\sqrt {2}}}}}$ Plotting in the cartesian system of (u,v) gives the folium rotated by 45° and therefore symmetric by u axis.

Since the folium is symmetrical about $y=x$ , it passes through the point $(3a/2,3a/2)$ .

## Relationship to the trisectrix of MacLaurin

The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation In geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742. In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

$x^{3}+y^{3}=3axy\,$ ,

and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting $x={{X+Y} \over {\sqrt {2}}},y={{X-Y} \over {\sqrt {2}}}$ . In the $X,Y$ plane the equation is

$2X(X^{2}+3Y^{2})=3{\sqrt {2}}a(X^{2}-Y^{2})$ .

If we stretch the curve in the $Y$ direction by a factor of ${\sqrt {3}}$ this becomes

$2X(X^{2}+Y^{2})=a{\sqrt {2}}(3X^{2}-Y^{2})$ which is the equation of the trisectrix of Maclaurin.

1. Simmons, p. 101
2. "DiffGeom3: Parametrized curves and algebraic curves". N J Wildberger, University of New South Wales . Retrieved 5 September 2013.