1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Parametrizing the folium of Descartes

  1. Aug 15, 2009 #1
    After recently studying Calc III on my own, I came across this problem with parametrizing the folium of Descartes.

    480px-Folium_Of_Descartes.svg.png

    1. The problem statement, all variables and given/known data

    Show that for t ≠ -1, 0, the line y = tx intersects the folium at the origin and at one other point P. Express the coordinates of P in terms of t to obtain a parametrization of the folium.

    2. Relevant equations

    The folium of Descartes is the curve with the equation x3 + y3 = 3axy, where a ≠ 0 is a constant.

    3. The attempt at a solution

    I couldn't really get anywhere with this problem. I understand why t ≠ -1, 0, and y = tx → t = y/x.
    Using the formula for the slope between two points with the origin (0, 0) and point P(x1, y1) on the folium went back to t = y/x

    I know that x(t) = 3at/(t3 + 1) and y(t) = 3at2/(t3 + 1) from the Internet but I couldn't find a good explanation of the steps in deriving the parametric equations. They look quite similar to the previous problem in the book which would probably help with this problem, but I couldn't do anything useful with that problem either:
    Show that the line of slope t through (-1, 0) intersects the unit circle in the point with coordinates x = (1-t2)/(t2 + 1), y = 2t/(t2 + 1)
    Conclude that these equations parametrize the unit circle with the point (-1, 0) excluded. Show further that t = y/(x + 1)
     
  2. jcsd
  3. Aug 15, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Bohrok! :smile:
    Just put t = y/x into x3 + y3 = 3axy :wink:
     
  4. Aug 15, 2009 #3
    Thanks so much tiny-tim! So annoying when I can't see the right path and start down a wrong one that doesn't work out... :redface:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Parametrizing the folium of Descartes
  1. Folium of Descartes (Replies: 1)

Loading...