X[s_, 
        t_] = (Sinh[2  s] Cos[2  t] - 
              2  Sinh[s] Sin[t])/(Cosh[s]^2  Cos[t]^2 + 
              Sinh[s]^2  Sin[t]^2) - s

Y[s_, 
        t_] = (-Cosh[2  s] Sin[2  t] + 
              2  Cosh[s] Cos[t])/(Cosh[s]^2  Cos[t]^2 + 
              Sinh[s]^2  Sin[t]^2) - t

curve[a_] = {X[s, a], Y[s, a]}

pic[0] = ParametricPlot[curve[0], {s, -10, 10}];

pic[a_] := 
        ParametricPlot[curve[a], {s, -10, 10}, 
          PlotRange -> {{-30, 30}, {-3, 3}}];

pic[ .1]

ParametricPlot[curve[0], {s, -20, 20}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .1], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .2], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .3], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .4], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .5], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .6], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .7], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .8], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[ .9], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.1], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.2], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.3], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.4], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.45], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.5], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}];

ParametricPlot[curve[1.51], {s, -10, 10}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 250];

ParametricPlot[curve[1.52], {s, -5, 5}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 500];

ParametricPlot[curve[1.52], {s, -2, 2}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.53], {s, -2, 2}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.57], {s, -2, 2}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[Pi/2], {s, -2, 2}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.6], {s, -2, 2}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.7], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.8], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[1.9], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.1], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.2], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.3], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.47], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-3, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.5], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.6], {s, -30, 30}, 
        PlotRange -> {{-30, 30}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.7], {s, -30, 30}, 
        PlotRange -> {{-10, 10}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.8], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.81], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.83], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.85], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.9], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[3], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-5, 5}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[3.1], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-7, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[3.14], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-7, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[3.15], {s, -30, 30}, 
        PlotRange -> {{-2, 2}, {-7, 3}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.8], {s, -5, 5}, 
        PlotRange -> {{-1, 1}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.76], {s, -5, 5}, 
        PlotRange -> {{-1, 1}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.78], {s, -5, 5}, 
        PlotRange -> {{-1, 1}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.79], {s, -5, 5}, 
        PlotRange -> {{-1, 1}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[2.795], {s, -5, 5}, 
        PlotRange -> {{-1, 1}, {-5, 0}}, 
        PlotPoints -> 1000];

ParametricPlot[curve[0], {s, -20, 20}, 
        PlotRange -> {{-5, 5}, {-3, 3}}];

ParametricPlot[curve[- .10], {s, -20, 20}, 
        PlotRange -> {{-5, 5}, {-3, 3}}];

ParametricPlot[curve[- .5], {s, -20, 20}, 
        PlotRange -> {{-5, 5}, {0, 10}}];

ParametricPlot[curve[-1], {s, -20, 20}, 
        PlotRange -> {{-5, 5}, {0, 10}}];

ParametricPlot[curve[-1.4], {s, -10, 10}, 
        PlotRange -> {{-5, 5}, {0, 30}}];

ParametricPlot[curve[-1.45], {s, -10, 10}, 
        PlotRange -> {{-5, 5}, {0, 50}}];

ParametricPlot[curve[-1.5], {s, -10, 10}, 
        PlotRange -> {{-5, 5}, {0, 100}}];

ParametricPlot[curve[-1.55], {s, -10, 10}, 
        PlotRange -> {{-5, 5}, {0, 100}}];

ParametricPlot[curve[-1.57], {s, -10, 10}, 
        PlotRange -> {{-5, 5}, {0, 500}}];