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
?
Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up.
Sign up to join this communityYou 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
.
Solve[{x == 2 Sin[t] + Sin[3 t], y == 2 Cos[t] - Cos[3 t]}, {t, y}]
. $\endgroup$y=f(x)
is not what I need. I needf(x, x^2, x^3, ..., y, y^2, y^3,...)=0
. :-) $\endgroup$