Parametric Plot where the parametric is solved from a equation

Question

I want to plot the image of

$ \qquad y-\sin t=\tan(2t)(x-\cos t) $

where the parameter $ t $ satisfies

$ \qquad\cos(t)(1+\sin^2(2t))-\sin(2t)\sin(t)=2x $.

In fact, I want to plot the envelope of

$ \qquad F=y-\sin(t)-\tan(2t)(x-\cos t)=0 \quad s.t.\quad\partial F/\partial t =0$

I use the command ContourPlot like so:

ContourPlot[{y - Sin[s] == Tan[2 s] (x - Cos[s]),
             Sin[s] (Cos[2 s]^2 - Cos[2 s] Sin[2 s]) + 2 Cos[s] == 2 x},
            {x, -1, 1}, {y, -1, 1}]

But it went wrong.

How can I get the correct image?

Answer 1

In fact, I want to plot the envelope of $ \qquad F=y-\sin(t)-\tan(2t)(x-\cos t)=0 \quad s.t.\quad\partial F/\partial t =0$

Here's how to quickly visualize an envelope of lines:

With[{n = 100}, 
     env = Graphics[{Directive[Opacity[1/2, Gray], Thin], 
                     Table[InfiniteLine[{Cos[t], Sin[t]}, {1, Tan[2 t]}],
                           {t, 0, 2 π, 2 π/(n - 1)}]}]]

Generate the parametric equations of the envelope:

Simplify[{x, Tan[2 t] (x - Cos[t]) + Sin[t]} /. 
First[Solve[D[Tan[2 t] (x - Cos[t]) + Sin[t], t] == 0, x]]]
   {(3 Cos[t] - Cos[3 t])/4, Sin[t]^3}

and show the lines and the curve together:

Show[env,
     ParametricPlot[{(3 Cos[t] - Cos[3 t])/4, Sin[t]^3}, {t, 0, 2 π}, PlotStyle -> Thick]]

---

As a bonus (and also as a demonstration on how to handle more complicated envelopes), here's another way to generate a nephroid, as the envelope of circles:

With[{circ = #.# &[{x, y} - {Sin[t]/2, Cos[t]/2}] - (Cos[t]/2)^2},
     neph = Simplify[First[GroebnerBasis[Append[{circ, D[circ, t]},
                                                Cos[t]^2 + Sin[t]^2 - 1],
                                         {x, y}, {Cos[t], Sin[t]}]]/y]]
   64 x^6 + 12 x^2 (1 - 4 y^2)^2 + 48 x^4 (-1 + 4 y^2) + (-1 + y^2) (1 + 8 y^2)^2

With[{n = 81},
     Show[Graphics[{Directive[Opacity[1/2, Gray], Thin], 
                    Table[Circle[{Sin[t]/2, Cos[t]/2}, Abs[Cos[t]/2]],
                          {t, 0, 2 π, 2 π/(n - 1)}]}, Frame -> True], 
          ContourPlot[neph == 0, {x, -3/4, 3/4}, {y, -1, 1}, ContourStyle -> Thick]]

Answer 2

I would do something very similar to @c186282's answer, except that I would solve things symbolically instead. For instance:

sol = Values @ First @ Solve[
    {
    y - Sin[t] == Tan[2t](x - Cos[t]),
    Cos[t] (1 + Sin[2t]^2) - Sin[2t] Sin[t] == 2x
    },
    {x, y}
]

{1/2 (Cos[t] - Sin[t] Sin[2 t] + Cos[t] Sin[2 t]^2), 1/2 (2 Sin[t] - Cos[t] Tan[2 t] - Sin[t] Sin[2 t] Tan[2 t] + Cos[t] Sin[2 t]^2 Tan[2 t])}

And then plot:

ParametricPlot[sol, {t, 0, 2Pi}, AspectRatio->1]

Notice the absence of the vertical artifacts at $x\approx \pm 0.35$. For you second question, I would proceed similarly:

F=y-Sin[t]-Tan[2t](x-Cos[t]) == 0;
sol = Values @ First @ Solve[{F, D[F, t]}, {x, y}]

ParametricPlot[sol, {t, 0, 2Pi}]

{1/2 (2 Cos[t] - Cos[t] Cos[2 t]^2 - Cos[2 t] Sin[t] Sin[2 t]), 1/2 (2 Sin[t] - Cos[t] Cos[2 t] Sin[2 t] - Sin[t] Sin[2 t]^2)}

Answer 3

First I make a function to solve for your $(x,y)$ point then I plot it.

Clear[xy]
xy[t_?NumericQ] := Module[{sol, x, y},
sol = Solve[{
  y - Sin[t] == Tan[2 t] (x - Cos[t]),
  Cos[t] (1 + Sin[2 t]^2) - Sin[2 t] Sin[t] == 2 x}, {x, y}];
Return[{x, y} /. sol[[1]]]
];

Notice I use ?NumericQ in the function definition this forces the numeric evaluation during the plotting.

Then I plot it:

ParametricPlot[xy[t], {t, 0, 2 Pi}, AspectRatio -> 1]