# Dictionary Definition

folium n : a thin layer or stratum of (especially
metamorphic) rock [also: folia (pl)]

# User Contributed Dictionary

## Latin

### Noun

- a leaf

#### Related terms

# Extensive Definition

## Definition

The Folium of Descartes is an algebraic
curve first proposed by Descartes in 1638
with an implicit equation:

- x^3 + y^3 - 3 x y = 0. \,

It can also be described explicitly in polar
coordinates as:

- r(\theta) = \frac.

## Characteristics of the curve

### Equation of the tangent

Using the method of implicit differentiation, we
can solve the above equation for y':

- \frac = \frac.

Using Point-slope form of the equation of a line,
we can find an equation to the tangent of the curve at (x_1 ,
y_1):

- y - y_1 = \frac(x - x_1).

### Horizontal and vertical tangents

The tangent line of Folium of Descartes is
horizontal when a y - x^2 = 0. Therefore, the tangent line is
horizontal when:

- x = a\sqrt[3].

The tangent line of Folium of Descartes is
vertical when y^2 - a x = 0. Therefore, the tangent line is
vertical when:

- y = a\sqrt[3].

This is explainable through a fact about the
symmetry of the curve. By looking at the graph, we can see that the
curve has two horizontal tangents and two vertical tangents. The
curve of Folium of Descartes is symmetrical about y = x, so if a
horizontal tangent has a coordinate of (x_1,y_1), there is a
corresponding vertical tangent, (y_1,x_1).

### Asymptote

The curve has an asymptote:

- x + y + a = 0.

The asymptote has a gradient of -1 and
x-intercept and y-intercept of -a.

## Algebraic components of the folium of Descartes

If we solve x^3 + y^3 = 3 a x y for y in terms of
x, we obtain the following three functions:

y = f(x) = \sqrt[3] + \sqrt[3]

and

y = \frac \left[ - f(x) \pm \sqrt \left( \sqrt[3]
- \sqrt[3] \right) \right ].

The reader may see that implicit differentiation
is a much easier way of obtaining an equation to the tangent of the
curve, rather than attempting to differentiate the above equations,
which is much more complicated than x^3 + y^3 = 3 a x y. As an
additional note, you may be able to see intuitively that it is
impossible to find a general formula for the roots of an nth-degree
equation, if n is any integer larger than 4.

folium in Bulgarian: Декартов лист

folium in German: Kartesisches Blatt

folium in Spanish: Folium de Descartes

folium in French: Folium de Descartes

folium in Dutch: Folium van Descartes

folium in Polish: Liść
Kartezjusza