documentclass[preview]{standalone}

usepackage{filecontents}

usepackage{tikz}
    usetikzlibrary{decorations.markings}
    usetikzlibrary{intersections}
begin{filecontents}{projection.tikz}
    % tangent: https://tex.stackexchange.com/a/25940/74942
    % intersection: https://tex.stackexchange.com/a/58216/74942
    begin{tikzpicture}[
    tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
            at position #1
            with
            {
                coordinate (tangent point-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                coordinate (tangent unit vector-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                coordinate (tangent orthogonal unit vector-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
            }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
    ] % tangent

    draw[help lines,xstep=.5,ystep=.5] (0,0) grid (4,3); %grid

    draw [
    name path=curve,
    tangent=0.4
    ] (0,0)
    to [out=60,in=180] (2,2)
    to [out=0, in=160] (3,1.75)
    to [out=-20, in=-120] (4,3);

    draw [blue, name path=tangent, use tangent] (-1,0) -- (2,0)
    node[sloped,inner sep=0cm,below,pos=.8,
    anchor=north west,
    minimum height=1cm,
    minimum width=1cm](N){a};

    path [name path=proj] (N.north west)% intersection
    node (P) {b} -- (N.south west);

    draw [name intersections={of=curve and proj}, blue] (P.center)
    --
    (intersection-1);

    end{tikzpicture}
end{filecontents}

begin{document}
input{./projection.tikz}
end{document}

documentclass[preview]{standalone}

usepackage{filecontents}

usepackage{tikz}
    usetikzlibrary{decorations.markings}
    usetikzlibrary{decorations.text}
    usetikzlibrary{intersections}
begin{filecontents}{projection.tikz}
    % tangent: https://tex.stackexchange.com/a/25940/74942
    % intersection: https://tex.stackexchange.com/a/58216/74942
    % text along path: https://tex.stackexchange.com/a/22316/74942
    begin{tikzpicture}[
    tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
            at position #1
            with
            {
                coordinate (tangent point-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                coordinate (tangent unit vector-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                coordinate (tangent orthogonal unit vector-pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
            }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
    ] % tangent

    draw[help lines,xstep=.5,ystep=.5] (0,0) grid (4,3); %grid

    defmyshift#1{raisebox{-2.5ex}} %text along path - height from line
    draw [
                name path=curve,
                tangent=0.4, % where tangent is set (percent of total curve)
                postaction={decorate,
                    decoration={text along path,
                        text align=center,
                        text={|sffamilyfootnotesizemyshift| and maybe here too }
                    }
                } %text command
                ] (0,0)
                    to [out=60,in=180] (2,2)
                    to [out=0, in=160] (3,1.75)
                    to [out=-20, in=-120] (4,3);

    defmyshift#1{raisebox{1ex}} %text along path - height from line
    draw [blue,
                name path=tangent,
                use tangent,
                postaction={decorate,
                    decoration={text along path,
                        text align=center,
                        text={|sffamilyfootnotesizemyshift| I want some text here }
                    }
                } %text command
                ] (-1,0) -- (2,0)
    node[sloped,inner sep=0cm,below,pos=.8,
    anchor=north west,
    minimum height=1cm,
    minimum width=1cm](N){a};

    path [name path=proj] (N.north west)% intersection
    node (P) {b} -- (N.south west);

    draw [ name intersections={of=curve and proj},
                blue,
                postaction={decorate,
                    decoration={text along path,
                        text align=center,
                        text={|sffamilytinymyshift| and here }
                    }
                } %text command
            ] (P.center)
    --
    (intersection-1);

    end{tikzpicture}
end{filecontents}

begin{document}
input{./projection.tikz}
end{document}