Folium of Descartes and Parametric EquationsDate: 11/23/98 at 01:12:20 From: Benjamin Goh Subject: Plotting an implicit function (e.g. y^3 + x^3 = 3xy) I have a mathematics dictionary at home, and I've come across a strange looking graph, called the folium of Descartes, with an equation of y^3 + x^3 = 3xy. After thinking about it, I realized that I can't simply create a table of x and y values to plot this curve, since I cannot make either y or x the subject of the equation. Many graph programs also are unable to plot this graph (Graphmatica, Equation Grapher. . .) I've found one way to plot such a graph, and that is testing every point on an x-y plane to see if it roughly fits the equation, but this method is rather slow. Is there another, better way to plot such implicit graphs? They are much more interesting than standard ones. I've been experimenting, and one of my favourites is tan(x^3) + log(sin(y^2)) = 3sin(x)cos(y). It has a very interesting pattern, but it's quite hard to plot. Date: 11/23/98 at 08:57:01 From: Doctor Jerry Subject: Re: Plotting an implicit function (e.g. y^3 + x^3 = 3xy) Hi Benjamin, Some equations can be described with parametric equations. It happens that the folium of Descartes can be so described. The equation x^3 + y^3 = 3*a*x*y is equivalent to: x = 3*a*t/(1+t^3) y = 3*a*t^2/(1+t^3) You can plot this by taking several values of t, calculating the x and y values, and plotting (x,y). I've plotted above equations using the Mathematica command: ParametricPlot[{3*t/(1+t^3),3*t^2/(1+t^3)},{t,-100,100}, PlotRange->{-1.5,2}] to get a rough sketch: The following the link, when available, will provide you with more information: http://mathworld.wolfram.com/FoliumofDescartes.html This link is from the CRC Encylopedia of Mathematics, by Eric Weisstein. He includes a sketch of the curve and says in part: [It is] a plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques. ... The curve has a discontinuity at t = -1. The left wing is generated as t runs from -1 to 0, the loop as t runs from 0 to infinity, and the right wing as t runs from -infinity to -1. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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