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### 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

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).

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

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