Tangent Line to the Lemniscate

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Warning: new definition for   norm
Warning: new definition for   trace
> interface(plotdevice=x11);

> # The lemniscate

> R := [sqrt(sin(2*t))*cos(t), sqrt(sin(2*t))*sin(t)];

                               1/2                 1/2
                 R := [sin(2 t)    cos(t), sin(2 t)    sin(t)]

> plot([R[1],R[2],t=0..2*Pi]);

> curve := plot([sqrt(sin(2*t))*cos(t), sqrt(sin(2*t))*sin(t),t=0..2*Pi]):

> # Position vector

> P := subs(t=Pi/4,R);

                          1/2                         1/2
         P := [sin(1/2 Pi)    cos(1/4 Pi), sin(1/2 Pi)    sin(1/4 Pi)]

> P := map(simplify,P);

                                      1/2       1/2
                           P := [1/2 2   , 1/2 2   ]

> # velocity vector

> v := subs(t=Pi/4, diff(R,t));

                cos(1/4 Pi) cos(1/2 Pi)              1/2
          v := [----------------------- - sin(1/2 Pi)    sin(1/4 Pi),
                                1/2
                     sin(1/2 Pi)

              sin(1/4 Pi) cos(1/2 Pi)              1/2
              ----------------------- + sin(1/2 Pi)    cos(1/4 Pi)]
                              1/2
                   sin(1/2 Pi)

> v := map(simplify,v);

                                       1/2       1/2
                          v := [- 1/2 2   , 1/2 2   ]

> # The tangent line
> L := evalm(P + s*v);
                         1/2          1/2       1/2          1/2
             L := [ 1/2 2    - 1/2 s 2   , 1/2 2    + 1/2 s 2    ]

> line := plot([L[1],L[2],s=-2..2]):

> with(plots): display3d({curve,line});