The Folium Of Descartes…
Today we are going to discuss about a unique curve or locus in mathematics. It’s named after one of the most revolutionary mathematicians of all time, Rene Descartes. Before I begin with the topic, I would like to give a brief introduction on the mathematician who had discovered this.
René Descartes (1596–1650) was a French mathematician, philosopher, and scientist often called the “Father of Modern Philosophy” and one of the founders of analytic geometry. He introduced the Cartesian coordinate system, which connected algebra with geometry and revolutionized mathematics. His famous statement, “Cogito, ergo sum” (“I think, therefore I am”), became a foundation of modern philosophy. Descartes made major contributions to geometry, optics, mechanics, and scientific reasoning, and his notable works include Discourse on the Method, Meditations on First Philosophy, Principles of Philosophy, and La Géométrie, where he developed coordinate geometry and studied important curves like the Folium of Descartes.
Now, let’s dive into today’s topic of discussion. The folium of Descartes is a special type of curve governed by the equation, x³+y³=3axy where a is any arbitrary constant. What’s so unique about this will be very clear once we have a look on its geometry. Here is the graph:
As you can see, the curve forms a loop (somewhat elliptical but in no sense an ellipse) at the origin. This is the only place in the entire 2 dimensional cartesian plane where the curve has an abnormal behaviour of forming a loop. In all other regions it just extends somewhat like a straight line. (Neglecting the initial regions on either side of the loop where its clearly following a curved trajectory).
This curve also has the special property of being symmetric about the line y=x.
This gives an important conclusion that the curve remains same even if we interchange the x and y coordinates in the original equation. Which is again a very noticeable fact for a mathematician by just looking at the equation due to the symmetry of the terms in the equation.
This curve has another beautiful property. If you notice the curve pretty closely at the origin, you can find that both the x axis and the y axis are 2 different tangents for the curve at a single point which is again the origin itself.
This curve also possess an oblique asymptote. An asymptote is a line that a curve approaches increasingly closely as the point on the curve moves toward infinity (or sometimes toward a particular point), but the curve does not actually meet the line in the limiting position. The equation of the asymptote is x+y+a=0. Note that the a involved over here is the same arbitrary constant which was part of the original equation of the folium of Descartes. The true geometric meaning of asymptote will be very clear if you could have a look on how it looks adjacent to the curve for which it is considered to be an asymptote.
As you can see, the line is approaching the curve and you might think that these are overlapping, but it clearly is not. You can check for yourself by just zooming in at the point you think they might coincide. They would never do. And that’s exactly why its an asymptote. This can also be verified by just simultaneously solving both the asymptote and the folium’s equation. There is actually only one value of a for which you get a solution and that is when a=0. But when this happens, the folium actually becomes a pair of straight lines both which are the same. So you get x+y=0 as the folium and the line so essentially it’s the same equation either ways.
The curve’s parametric coordinates can be given as follows:
These becomes very essential in tracing the curve. And this is one of the best ways to obtain the area enclosed by the cute loop in this. Let’s get into areas now…
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There are mainly 2 things which I would be discussing under this section. One being the actual derivation of the area enclosed by the loop. Purely calculus oriented. And the second one being a property whose proof is beyond the scope of this post to cover at this point of time.
Let me discuss the property first. You see, this is one of the most craziest features of this curve according to me. The area enclosed by the loop and the area enclosed between the asymptote and the curve are exactly equal. The value? Well, that we’ll surely get through the derivation which I'll be performing next but this is surely crazy since the area enclosed by a finite part of the curve is same as that of the area enclosed between a line and the same curve from negative infinity to positive infinity.
Now let’s derive the loop’s area. Graphically speaking, this region over here:
We already know from the parametric coordinates of the equation that
If we differentiate this with respect to t, we get:
To obtain the area we need to compute the integral:
We need to substitute the parametric coordinate of y in terms of t and replace dx in terms of t and dt from dx/dt which we obtained previously. Ultimately we get to solve this integral over here:
Just be making a u substitution for t³ or 1+t³ it becomes an elementary integral which gives us with the final answer of A=3a²/2. I am skipping the part of u substitution and evaluation of the integral since that can be done by anyone who has understood all of the things mentioned in this post.
I particularly found the derivation of area of the loop in this really interesting since it was my first time of replacing both y and dx in terms of a completely new variable ( t in this case). I am not much familiar with this method of integrating but this really is a very clever approach towards the problem of finding out the area.
I hope you enjoyed reading this.
Thank You!…
