The Archimedes spiral equation in parametric form:
fx[t_] := Cos[t]*t;
fy[t_] := Sin[t]*t;
How can it be converted to implicit form?
Is there a general method to convert any parametric equation to implicit form?
fx[t_] := Cos[t]*t;
fy[t_] := Sin[t]*t;
ParametricPlot[{fx[t], fy[t]}, {t, 0, 10}]
sol = Assuming[Element[{x, y}, Reals],
Eliminate[{x == fx[t], y == fy[t]}, t,
InverseFunctions -> True] //
FullSimplify]
t^2 == x^2 + y^2 && x == t Cos[t]
Since t
was not eliminated
solxy = sol[[-1]] /. Solve[sol[[1]], t, Reals] // FullSimplify
(* {x + Sqrt[x^2 + y^2] Cos[Sqrt[x^2 + y^2]] == 0,
x == Sqrt[x^2 + y^2] Cos[Sqrt[x^2 + y^2]]} *)
The result is the original spiral and its mirror
ContourPlot[Evaluate@solxy,
{x, -10, 10}, {y, -10, 10},
AspectRatio -> 1,
PlotPoints -> 100,
MaxRecursion -> 5]
x Tan[Sqrt[x^2 + y^2]] == y
$\endgroup${x == Exp[t] + t^2, y == t - Cos[t]}
. $\endgroup$