Folium of Descartes - Graphing The Curve

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:

which can be plotted easily. Another technique is to write y = px and solve for x and y in terms of p. This yields the parametric equations:

.

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:

and yields

Plotting in the cartesian system of (u,v) gives the folium rotated by 45° and therefore symmetric by u axis.