Parabola Parallels

1. Normal: \[ \underline{n} = \frac{1}{\sqrt{1+a^2 t^2}} \begin{pmatrix} -at\\1 \end{pmatrix} \]
2. Evolute: \[ \underline{c}(t) = \begin{pmatrix} -t^3\\\frac{3}{2}t^2+1 \end{pmatrix} \]
3. Theorem: The Family of Parallel Curves has the same Evolute.
4. Theorem: Singular Points on a Parallel Curve Coincides with the Evolute Point

\input{../../../Draw.Curve.tikz}


%{\t}
\newcommand{\R}[1]
{
   \tikzmath{\t=#1;}
   \coordinate (R) at (\t,0.5*\t*\t);
}

%{\t}: velocity
\newcommand{\V}[1]
{
   \tikzmath{\t=#1;}
   \tikzmath{\v=sqrt(1+\t*\t);}
}

\newcommand{\Normal}[1]
{
   \tikzmath{\t=#1;}
   \V{\t}
   \tikzmath{\vv=1/\v;}
   \coordinate (N) at (-\vv*\t,\vv);
}


\newcommand{\Evolute}[1]
{
   \tikzmath{\t=#1;}
   \coordinate (C) at (-\t*\t*\t,3/2*\t*\t+1);
}




\newcommand{\Parallel}[2]
{
   \tikzmath{\d=#1;}
   \tikzmath{\t=#2;}
   \R{\t}
   \Normal{\t}
   \coordinate (P) at ($(R)+\d*(N)$);
}


\tikzmath{\NN=10;}
\foreach \n in {1,2,...,\NN}
{
   \tikzmath{\d=3*\n/\NN;}
   %Convex combination of colors
   \tikzmath{\dc=100*\n/\NN;}
   \def\col{blue!\dc!cyan}

   \DrawParallel{-3}{3}{200}{\d}{\col}
}

\DrawParallel{-3}{3}{200}{1}{cyan,thick}
\DrawCurve{-3}{3}{100}{blue}
\DrawEvolute{-1.5}{1.5}{100}{orange}