TiKz-pgf: Project intersection points in the abscissa axis

Question

With some effort, I was able to find a way to mark the intersection points between the Gaussian bell and the straight line. Leaving aside the first point, I would need to project the two intersection points onto the abscissa axis. My idea is this: extract the x coordinate of the intersection point, called e.g. x_1, and create a node at point (x_1,0).

Is it possible to accomplish such a thing?

Here my MWE

\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\pgfmathdeclarefunction{gauss}{2}{%
  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
\newcommand*{\ShowIntersection}[2]{
\fill 
    [name intersections={of=#1 and #2, name=i, total=\t}] 
    [red, opacity=1, every node/.style={above left, black, opacity=1}] 
    \foreach \s in {1,...,\t}{(i-\s) circle (2pt)
        node [above left] {\s}};
}
\begin{document}

\begin{figure}
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            yticklabels=\empty, xticklabels=\empty,
            width=8cm,
            height=6cm,
            domain=-2:3, ymin = 0.07, xmin = -2, enlargelimits=upper,
            xtick=\empty, ytick=\empty,
            clip mode=individual,
            xlabel={$T$}, ylabel={$\Delta T$},
            ]
            \addplot[smooth,red,samples=50,domain=-2:3,name path=gaussian]{8 * gauss(1,0.75)};
            \addplot[smooth,blue,name path=line]{x*1.13 + 1.36};
            \ShowIntersection{gaussian}{line}
        \end{axis}
    \end{tikzpicture}
\end{figure}

\end{document}

Answer 1

This solution draws the projection of the intercepts onto the x-axis and retrieves the value x using \pgfplotspointgetcoordinates

\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{intersections}

\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\pgfmathdeclarefunction{gauss}{2}{%
    \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
    
\newcommand*{\ShowIntersection}[2]{
\draw [
name intersections={of=#1 and #2,name=i, total=\t}, thin, dotted] 
[every node/.style={black,font=\tiny}]  
\foreach \s in {1,...,\t}
{(i-\s)  node[above left] {\s} % dysplay number of the intersection
    node[fill,red,circle,inner sep=0,minimum size=4pt]{} % big dot in the intercection
    (i-\s |- i-\s) -- (i-\s |- current axis.south) % line to projet from intesection to x-axis
    node [below,font=\tiny] {\pgfplotspointgetcoordinates{(i-\s)} % recover x-y coordinates of the intesection
        $\pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/x}}$} % and put x below the line    
};
}

\begin{document}
    
\begin{figure}
        \centering
\begin{tikzpicture}
\begin{axis}[
    yticklabels=\empty, xticklabels=\empty,
    width=8cm,
    height=6cm,
    domain=-2:3, ymin = 0.07, xmin = -2, enlargelimits=upper,
    xtick=\empty, ytick=\empty,
    clip mode=individual,
    xlabel={$T$}, ylabel={$\Delta T$},
    ]
    \addplot[smooth,red,samples=50,domain=-2:3,name path=gaussian]{8 * gauss(1,0.75)};
    \addplot[smooth,blue,name path=line]{x*1.13 + 1.36};
    \ShowIntersection{gaussian}{line}
\end{axis}
\end{tikzpicture}
\end{figure}
    
\end{document}

Now it's easy to add the y-intercepts

\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{intersections}

\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\pgfmathdeclarefunction{gauss}{2}{%
    \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
    
\newcommand*{\ShowIntersection}[2]{
\draw [
name intersections={of=#1 and #2,name=i, total=\t}, thin, dotted] 
[every node/.style={black,font=\tiny}]  
\foreach \s in {1,...,\t}
{(i-\s)  node[above left] {\s} % dysplay number of the intersection
    node[fill,red,circle,inner sep=0,minimum size=4pt]{} % big dot in the intercection
    (i-\s |- i-\s) -- (i-\s |- current axis.south) % line to projet from intesection to x-axis
    node [below,font=\tiny] {\pgfplotspointgetcoordinates{(i-\s)} % recover x-y coordinates of the intesection
        $\pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/x}}$} % and put x below the line
    ( i-\s|- i-\s)--    (current axis.west |- i-\s)% line to projet from intesection to y-axis
    node [left,font=\tiny] {\pgfplotspointgetcoordinates{(i-\s)} % do it again
        $\pgfmathprintnumber[fixed]{\pgfkeysvalueof{/data point/y}}$}   % and put y left of the line
};
}

\begin{document}
    
\begin{figure}
        \centering
\begin{tikzpicture}
\begin{axis}[
    yticklabels=\empty, xticklabels=\empty,
    width=8cm,
    height=6cm,
    domain=-2:3, ymin = 0.07, xmin = -2, enlargelimits=upper,
    xtick=\empty, ytick=\empty,
    clip mode=individual,
    xlabel={$T$}, ylabel={$\Delta T$},
    ]
    \addplot[smooth,red,samples=50,domain=-2:3,name path=gaussian]{8 * gauss(1,0.75)};
    \addplot[smooth,blue,name path=line]{x*1.13 + 1.36};
    \ShowIntersection{gaussian}{line}
\end{axis}
\end{tikzpicture}
\end{figure}
    
\end{document}