# Tangent Line to the Lemniscate

```    |\^/|     Maple V Release 3 (SUNY at Albany)
._|\|   |/|_. Copyright (c) 1981-1994 by Waterloo Maple Software and the
<____ ____>  are registered trademarks of Waterloo Maple Software.
|       Type ? for help.
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});
```

Carlos Rodriguez <carlos@math.albany.edu>