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.
Read more about this topic: Folium Of Descartes
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