Is it possible to draw an ellipse with perscribed major or minor axis?

Question

Suppose we programmatically calculate two coordinates P and Q in TikZ and then choose a positive number r. This data defines a unique ellipse:

The center of this ellipse is the midpoint between P and Q and the value of R is half the distance between P and Q.

My questions are:

Question 1. Is there a simple way to draw this ellipse in TikZ?

Question 2. More generally, is there a simple way to draw arbitrary arcs on this ellipse?

Answer 1

While you are waiting for the TikZ helpers, here is a Metapost version. This is almost a one-liner in MP.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}

vardef elliptic_path(expr a, b, minor_radius) =
    fullcircle xscaled abs(a-b) yscaled 2 minor_radius
               rotated angle (a-b) shifted 1/2[a,b]
enddef;

beginfig(1);
    pair P, Q; numeric r;
    P = origin; Q = 89 dir 42;
    r = 34;

    path e; 
    e = elliptic_path(P, Q, r);

    draw e withpen pencircle scaled 1.414 withcolor 3/4 white;
    drawarrow subpath (1, 5) of e withcolor red;

    dotlabel.lft("$P$", P);
    dotlabel.rt("$Q$", Q);
endfig;
\end{mplibcode}
\end{document}

This is wrapped up in luamplib so compile with the lualatex engine.

Answer 2

I think this is a bit of an xy question. What you probably want to have on the long run is something that transforms you in a coordinate system the center of which is the center between two points, the x axis goes from the center to one of the two points, and the y axis is orthogonal and has length r. Then a circle in this coordinate system becomes your ellipse, ordinary arcs, elliptical arcs, and so on. Obviously, TikZ does that out of the box. To make things a bit more convenient I added a style, elli cs, that installs this coordinate system from these data. With this style, the graph becomes as simple as

 \begin{scope}[elli cs={A={(P)},B={(Q)},r=9mm}]
  \draw circle[radius=1];
  \draw[dashed] (0,1) -- node[midway,fill=white]{$r$} (0,0) node[dot] {}
  -- node[midway,fill=white]{$R$} (1,0);
  \draw[blue,-{Stealth[bend]}] (-30:1) arc[start angle=-30,end angle=120,radius=1];
 \end{scope}

As you see, for the arc there is no guess work required, the angles have a very intuitive interpretation in this coordinate system. You can do many further operations in this frame very conveniently, and also transform shapes according to it.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{arrows.meta,bending,calc}
\begin{document}
\begin{tikzpicture}[dot/.style={circle,fill,inner sep=1pt},thick,
    elli cs/.code={
    \tikzset{ellipse through/.cd,#1}
    \def\pv##1{\pgfkeysvalueof{/tikz/ellipse through/##1}}%
    \edef\temp{\noexpand\tikzset{shift={($0.5*\pv{A}+0.5*\pv{B}$)},
        x={($0.5*\pv{B}-0.5*\pv{A}$)},
        y={($($\pv{A}!\pv{r}!90:\pv{B}$)-\pv{A}$)}
        }}%
    \temp   
    },
    ellipse through/.cd,r/.initial=5mm,A/.initial={(-1,0)},
        B/.initial={(1,0)}]
   \path[nodes=dot] (0,0) node[label=below left:$P$] (P){}
     (4,1.5) node[label=below right:$Q$] (Q){};         
   \begin{scope}[elli cs={A={(P)},B={(Q)},r=9mm}]
    \draw circle[radius=1];
    \draw[dashed] (0,1) -- node[midway,fill=white]{$r$} (0,0) node[dot] {}
    -- node[midway,fill=white]{$R$} (1,0);
    \draw[blue,-{Stealth[bend]}] (-30:1) arc[start angle=-30,end angle=120,radius=1];
   \end{scope}
\end{tikzpicture}
\end{document}