# Elimination of parameter

1. Mar 16, 2009

### hhn2002

basically i need to convert the following parametric equation:
x=a cos (T) + b cos (T/2)
y=a sin (T) + b sin (T/2); where a and b are coefficients

to cartesian form: y = f(x)

i need it in that form so i can do calculus analysis for an improved engine design i hope to develop. thanks in advance for any help.

ps. if it helps any its a graph of an epitrochoid.
so ive read that it may be impossible to solve but free brownie points to those that can!

Last edited: Mar 16, 2009
2. Mar 16, 2009

### John Creighto

Write the trigonometric functions as exponential functions using euler's identity. Replace exp(iT/2) with z. Multiply though by z^2. Solve the fourth order polynomial symbolically (Yicks!) for z. T=2*angle(z). This will give you Z in terms of x and then you can then substitute this result into the seconded equation to get y in terms of x.

It's going to be extremely messing because the symbolic solution of a forth order polynomial is not simple. I'd give it a try if you pay me or find me a job. Best of luck :)

P.S. That's a cool looking shape.
http://en.wikipedia.org/wiki/Epitrochoid

Last edited: Mar 16, 2009