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Why I am only getting one point of intersection read by the program? Solution is to breaking to two parts with tangent lines drawn at different lengths.

documentclass{article}

usepackage{tikz}

usetikzlibrary{calc,intersections}

usetikzlibrary{decorations.markings}

begin{document}

pgfkeys{tikz/.cd,

 tangent length/.store in=TangentLength,

 tangent length=3.14cm,

 normal length/.store in=NormalLength,

 normal length=7mm}

 newcounter{tangent}

 newcounter{normal}

 tikzset{tangent/.style={red,thin},normal/.style={blue,thin},

 tangent at/.style={postaction={decorate,decoration={markings,

 mark=at position #1 with {stepcounter{tangent}

 draw[tangent,name path=tangent-thetangent] (-TangentLength,0) -- 

 (9cm,0);

 fill[tangent] (0,0) circle (2pt);}}}},

  normal at/.style={postaction={decorate,decoration={markings,

   mark=at position #1 with {stepcounter{normal}

 draw[normal,name path=normal-thenormal] (0,-NormalLength) -- 

 (0,NormalLength);

 fill[normal] (0,0) circle (2pt);}}}}, 

 }

 begin{tikzpicture}

 % Axes

 draw [-latex] (-1,0) -- (11,0) node [right] {$x$};

 draw [-latex] (0,-1) -- (0,6) node [above] {$y$};

 % Origin

 node at (0,0) [below left] {$0$};

 % Points

 coordinate (start) at (1,-0.8);

 coordinate (c1) at (3,3);

 coordinate (c2) at (5.5,1.5);

 coordinate (c3) at (8,4);

 coordinate (end) at (10.5,-0.8);

 % show the points

 % foreach n in {start,c1,c2,c3,end} fill [black] (n)

 %     circle (2pt) node [below] {}; 

 %  join the coordinates

draw [name path=curve,thick,tangent at/.list={0.3,0.3},

  normal at/.list={}] (start) to[out=70,in=180] (c1) to[out=0,in=180]

  (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);

 fill[name intersections={of=curve and tangent-2,total=t},cyan]

 (intersection-t) circle (2pt);

  fill[name intersections={of=curve and tangent-1,total=t},cyan]

  (intersection-t) circle (2pt);

 end{tikzpcicture}

 end{document}

This outputs:

The style tangent at=<pos> attaches a tangent at position pos. So tangent at/.list={0.3,0.3} will just create two identical tangents. However, this has nothing to do with the intersections. Rather, you want to mark the last two intersections of the tangent with the curve. (TikZ finds more intersections since a tangent of course runs very close to the curve, so some of the additional intersections are not true intersections but result from numerical inaccuracies.)

documentclass{article}

usepackage{tikz}

usetikzlibrary{calc,intersections}

usetikzlibrary{decorations.markings}

begin{document}

pgfkeys{tikz/.cd,

 tangent length/.store in=TangentLength,

 tangent length=3.14cm,

 normal length/.store in=NormalLength,

 normal length=7mm}

 newcounter{tangent}

 newcounter{normal}

 tikzset{tangent/.style={red,thin},normal/.style={blue,thin},

 tangent at/.style={postaction={decorate,decoration={markings,

 mark=at position #1 with {stepcounter{tangent}

 draw[tangent,name path=tangent-thetangent] (-TangentLength,0) -- 

 (9cm,0);

 fill[tangent] (0,0) circle (2pt);}}}},

  normal at/.style={postaction={decorate,decoration={markings,

   mark=at position #1 with {stepcounter{normal}

 draw[normal,name path=normal-thenormal] (0,-NormalLength) -- 

 (0,NormalLength);

 fill[normal] (0,0) circle (2pt);}}}}, 

 }

 begin{tikzpicture}

 % Axes

 draw [-latex] (-1,0) -- (11,0) node [right] {$x$};

 draw [-latex] (0,-1) -- (0,6) node [above] {$y$};

 % Origin

 node at (0,0) [below left] {$0$};

 % Points

 coordinate (start) at (1,-0.8);

 coordinate (c1) at (3,3);

 coordinate (c2) at (5.5,1.5);

 coordinate (c3) at (8,4);

 coordinate (end) at (10.5,-0.8);

 % show the points

 % foreach n in {start,c1,c2,c3,end} fill [black] (n)

 %     circle (2pt) node [below] {}; 

 %  join the coordinates

draw [name path=curve,thick,tangent at=0.3] (start) to[out=70,in=180] (c1) to[out=0,in=180]

  (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);

 fill[name intersections={of=curve and tangent-1,total=t},cyan]

 pgfextra{pgfmathtruncatemacro{prevt}{t-1}}

 foreach i in {prevt,t} {(intersection-i) circle (2pt)};

end{tikzpicture}

end{document}