Tikz: path generating a large blank space

Question

Let's suppose the following situation (wich is the simplest I could think): a circle intersepts two different points of a line and we call these points C and D.

\documentclass[a4paper, 10pt]{article}
    \usepackage{tikz}
    \usetikzlibrary{intersections}
\begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    \begin{figure}[h!]
    \centering
        \begin{tikzpicture}
        \coordinate (A) at (-2,0);
        \coordinate (B) at (2,0);
        \coordinate (P) at (0,1);
        \draw [name path=line] (A)--(B);
        \draw [name path=circle] (P) circle (1.5cm);
        \path [name intersections ={of=line and circle,name=N,total=\t}];
        \node at (N-1)[below left]{C};
        \node at (N-2)[below right]{D};
        \end{tikzpicture}
    \end{figure}

Which lead us to the meaning of the figure.
\end{document}

My situation actually uses the through library, would be, in the example, something like giving a point (C or D) of the line and drawing a circle from P passing through this point. I don't really know if it does change the problem. The thing is: what if I just wanted the coordinates? Instead of using \draw, I'd use \path (or \node at (?) [circle through (...)];) and whatever commands to simply get the coordinates. In the example, I just switched \draw [name path=circle] (P) circle (1.5cm); to \path [name path=circle] (P) circle (1.5cm);

\documentclass[a4paper, 10pt]{article}
    \usepackage{tikz}
    \usetikzlibrary{intersections}
\begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    \begin{figure}[h!]
    \centering
        \begin{tikzpicture}
        \coordinate (A) at (-2,0);
        \coordinate (B) at (2,0);
        \coordinate (P) at (0,1);
        \draw [name path=line] (A)--(B);
        \path [name path=circle] (P) circle (1.5cm);
        \path [name intersections ={of=line and circle,name=N,total=\t}];
        \node at (N-1)[below left]{C};
        \node at (N-2)[below right]{D};
        \end{tikzpicture}
    \end{figure}

Which lead us to the meaning of the figure.
\end{document}

This huge blank space. How to get rid of it in order to get something like this:

By the way, for the last image I just typed C and D coordinates as approximations of the actual X coordinate for both of them.

Answer 1

Even if paths are not drawn, they still contribute to the bounding box. So you may want to use the overlay key on auxiliary paths and coordinates. overlay makes sure that they do not contribute to the bounding box.

\documentclass[a4paper, 10pt]{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    \begin{figure}[h!]
    \centering
        \begin{tikzpicture}
        \path (-2,0) coordinate (A) (2,0) coordinate (B);
        \path[overlay] (0,1) coordinate (P);
        \draw [name path=line] (A)--(B);
        \path[overlay,name path=circle] (P) circle[radius=1.5cm];
        \path [name intersections ={of=line and circle,name=N}];
        \node at (N-1)[below left]{C};
        \node at (N-2)[below right]{D};
        \end{tikzpicture}
    \end{figure}

Which lead us to the meaning of the figure.
\end{document}

Answer 2

Another approach is to reset the bounding box and recreate it to include only specific nodes and scopes (using local bounding boxes).

\documentclass[a4paper, 10pt]{article}
    \usepackage{tikz}
    \usetikzlibrary{intersections,fit}
\begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    \begin{figure}[h!]
    \centering
        \begin{tikzpicture}
        \coordinate (A) at (-2,0);
        \coordinate (B) at (2,0);
        \coordinate (P) at (0,1);
        \draw [name path=line] (A)--(B);
        \path [name path=circle] (P) circle (1.5cm);
        \path [name intersections ={of=line and circle,name=N,total=\t}];
        \node (C) at (N-1)[below left]{C};
        \node (D) at (N-2)[below right]{D};
        \pgfresetboundingbox
        \node[fit=(A) (B) (C) (D)] {};
        \draw[red] (current bounding box.south west) rectangle (current bounding box.north east);
        \end{tikzpicture}
    \end{figure}

Which lead us to the meaning of the figure.
\end{document}