Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Deriving Parametric Equations for Cissoid of Diocles

Date: 02/12/2005 at 23:12:31
From: Amy
Subject: Parametric Equations

How do you derive the parametric equations for the Cissoid of Diocles?
So few variables are given in the problem, and using just cosine or
sine won't help you get it.  There's a combination.



Date: 02/13/2005 at 11:46:42
From: Doctor Jerry
Subject: Re: Parametric Equations

Hello Amy,

See this figure:

  http://mathforum.org/dr.math/gifs/cissoid.jpg 

Here's a derivation:

Using the above figure, we define the cissoid by choosing an arbitrary
point S with coordinates (2a,q) on the line x = 2a.  We find the point
of intersection of the line OS with the circle (x-a)^2 + y^2 = a^2 to be

  ( 8a^3/(4a^2 + q^2), 4a^2q/(4a^2 + q^2) ).

Corresponding to the point S, we define P as the point on the line
from O to S for which |OP| = |RS|.  With a bit of algebra, we find the
coordinates of P to be

  x = 2aq^2/(q^2+4a^2)

  y = q^3/(q^2+4a^2).

If one eliminates q from these equations, one finds 

  x^3 + x*y^2 = 2*a*y^2.

If one lets q = 2a*tan(t), motivated by the q^2 + 4a^2 in the 
denominator, one finds

  x = 2a*sin^2(t)

  y = 2a*sin^3(t)/cos(t).

If you have any questions about my comments, please write back.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Euclidean Geometry

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/