4
$\begingroup$

I have a parametric curve defined by

x = 2 Sin[t] + Sin[3 t]   
y = 2 Cos[t] - Cos[3 t]

How can I eliminate t so get a single implicit equation in x and y?

Improve this question
$\endgroup$
3
  • $\begingroup$ Non-implicit solution: Solve[{x == 2 Sin[t] + Sin[3 t], y == 2 Cos[t] - Cos[3 t]}, {t, y}]. $\endgroup$
    – T.F
    Jul 24, 2017 at 5:20
  • $\begingroup$ have you tried the code? When you solve for y,t you automatically eliminate t... Hence the equation is expressed in terms of x and y. Next time try running the code... $\endgroup$
    – T.F
    Jul 24, 2017 at 6:18
  • $\begingroup$ @T.F: Sorry. y=f(x) is not what I need. I need f(x, x^2, x^3, ..., y, y^2, y^3,...)=0. :-) $\endgroup$
    – Second Person Shooter
    Jul 24, 2017 at 6:49

1 Answer 1

Reset to default
5
$\begingroup$

You should first rationalize the trig functions and the use Eliminate:

Clear[x, y]

Eliminate[
 TrigExpand[{x == 2 Sin[t] + Sin[3 t], y == 2 Cos[t] - Cos[3 t]} /. 
   t -> 2 ArcTan[u]], u]

(*
==> x^6 + x^4 (17 + 3 y^2) + x^2 (63 - 94 y^2 + 3 y^4) == 
 81 - 63 y^2 - 17 y^4 - y^6
*)

The replacement rule introduces the rationalization with variable u.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.