Plot a parametric curve subject to some condition

Question

I have a parametric curve described by $(x(s,t), y(s,t))$ where I would like to consider $-1\leqslant s,t \leqslant 1$ subject to some additional constraint, say $f(s,t) = 0$. I know I could solve for $s$ in terms of $t$ and plot for the specified $t$ values, but $f(s,t)$ is itself an implicit function of $s$ and $t$ and solving for $s$ would involve splitting into many different cases. Is there anyway to do this directly in Mathematica?

I tried

ParametricPlot[{x,y},{s,-1,1},{t,-1,1},
    RegionFunction->Function[{x,y,s,t}, f[s,t]==0]
]

But this returns an empty plot and I'm not confident my syntax is correct as I'm quite new to Mathematica.

Answer 1

Since you didn't define x[s, t], y[s, t] and ff[s, t], I will contrive a simple example by defining

x[s_, t_] := Cos[π s] Cos[π t]
y[s_, t_] := Sin[π s] Cos[π t]
f[s_, t_] := Sin[π (s + t)] - Cos[π s t]

With these definitions

p1 =
  ParametricPlot[{x[s, t], y[s, t]}, {s, -1, 1}, {t, -1, 1},
    PlotStyle -> Lighter[Red, .5]]

gives a circular region, but when the option RegionFunction -> (f[#3, #4] == 0 &), the plot is empty because the righthand side of the rule is a contour line, not a region.

However, when I make the righthand side of the rule a real region specification, something which is perhaps interesting is plotted.

p2 = 
  ParametricPlot[{x[s, t], y[s, t]}, {s, -1, 1}, {t, -1, 1},
    PlotPoints -> 200,
    PlotRange -> {{-1, 1}, {-1, 1}},
    RegionFunction -> (Abs @ f[#3, #4] < .003 &)];

Show[p1, p2]