Filling between an ellipse and a line

Question

I want to shade from the line QV to the edge of the ellipse. How can I do this?

\documentclass[11pt]{article}
\usepackage{amsmath, amssymb, eucal, yfonts, setspace, sectsty, enumitem, amscd, caption, tikz, tikz-qtree, mathtools, inconsolata, pgfplots, tikz-3dplot, pxfonts, xcolor}
\usepackage[margin=0.75in]{geometry}

\usetikzlibrary{arrows,decorations.markings,calc,fadings,decorations.pathreplacing, patterns, decorations.pathmorphing, positioning}
\allsectionsfont{\sffamily\raggedright\underline}
\begin{document}

\begin{center}
\begin{tikzpicture}
\draw (-3.25,0) -- (5,0) node[scale = .8, fill = white] at (-1.625,0) {$a$};
\draw (0,-3.25) -- (0,3.25) node[scale = .8, fill = white] at (0,1.25) {$b$};
\draw[red] (0,0) circle (3cm);
\draw[blue] (0,0) ellipse (3cm and 2.5cm);
\draw (0,0) -- (2.59808,1.5) node[scale = .8, fill = white] at (1.29904,0.75) {$a$};
\filldraw (-3,0) circle (.07cm) node[scale = .8] at (-3.2,-0.25) {$A$};
\filldraw (3,0) circle (.07cm) node[scale = .8] at (3.2,-0.25) {$P$};
\filldraw (0,-2.5) circle (.07cm) node[scale = .8] at (-0.25,-2.7) {$D$};
\filldraw (0,2.5) circle (.07cm) node[scale = .8] at (-0.25,2.7) {$B$};
\filldraw (1.75,0) circle (.07cm) node[scale = .8] at (1.5,-0.25) {$F$};
\filldraw (2.59808,1.5) circle (.07cm) node[scale = .8] at (2.84808,1.75) {$Q$}; 
\filldraw (0,0) circle (.07cm) node[scale = .8] at (-0.25,-0.25) {$O$};
\draw (2.59808,1.5) -- (2.59808,0);
\filldraw (2.59808,0) circle (.07cm) node[scale = .8] at (2.34808,-0.25) {$V$};
\filldraw[inner color = pink!70!, outer color = blue!40!black] (0,0) -- (0.875,0) arc (0:30:0.875cm);
\node[scale = .8, fill = white] at (0.875,0) {$ae$};
\node[scale = .8] at (1,0.28) {$E$};
\end{tikzpicture}
\end{center}
\end{document}

Answer 1

You only need to clip the ellipse. You can get some coordinate with intersections library if you want.

\documentclass[11pt]{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}

\draw[red] (0,0) circle (3cm);
\draw[blue] (0,0) ellipse (3cm and 2.5cm); 
\begin{scope}
    \clip (0,0) ellipse (3cm and 2.5cm); 
    \fill[blue!15] (2.59808,1.5) rectangle (3,0);  
\end{scope}
\draw (2.59808,1.5) -- (2.59808,0);  
\draw (-3.25,0) -- (5,0) node[scale = .8, fill = white] at (-1.625,0) {$a$};
\draw (0,-3.25) -- (0,3.25) node[scale = .8, fill = white] at (0,1.25) {$b$};   

\filldraw (3,0) circle (.07cm) node[scale = .8] at (3.2,-0.25) {$P$};
\filldraw (2.59808,1.5) circle (.07cm) node[scale = .8] at (2.84808,1.75) {$Q$}; 
\filldraw (2.59808,0) circle (.07cm) node[scale = .8] at (2.34808,-0.25) {$V$};

\end{tikzpicture}    
\end{document}

Answer 2

You can use the intersections library to find the coordinate where the ellipse and the line from V to Q intersect.

1. Name the paths you want to intersect:

   \draw[name path=ellipse, blue] (0,0) ellipse (3cm and 2.5cm);
   \draw[name path=line VQ] (2.59808,1.5) -- (2.59808,0);
   

2. Use the name intersections key to find the intersection. (It will be available as (Q'-1), in case of more than one intersection, TikZ append -<number of intersection> at the name.)

   name intersections={of=line VQ and ellipse, name=Q'}
   

3. Calculate the angle of that point with the calc library. (As the center of the ellipse is (0,0) we get directly the angle from the center.)

   let \p{Q'} = (Q'-1),
       \n{Q'} = {atan2(\x{Q'}/3cm,\y{Q'}/2.5cm)}
   in
   

   In the following path you can use \n{Q'} for the angle of the intersection point.

4. Draw/fill/pattern/… the path between V, P and the intersection Q'.

   (2.59808,0) -- (3cm,0cm)                                                 % from V to P
   arc[x radius = 3cm, y radius = 2.5cm, start angle = 0, end angle=\n{Q'}] % from P to Q'
   -- cycle;                                                                % back to V
   

Code

Or in one entire path:

% Preamble:
\usetikzlibrary{intersections}

% TikZ picture:
\draw[name path=ellipse, blue] (0,0) ellipse (3cm and 2.5cm);
\draw[name path=line VQ] (2.59808,1.5) -- (2.59808,0);

\path[name intersections={of=line VQ and ellipse, name=Q'}]
    let \p{Q'} = (Q'-1),
        \n{Q'} = {atan2(\x{Q'}/3cm,\y{Q'}/2.5cm)}
    in [
        draw=green,
        fill=green!50
    ] (2.59808,0) -- (3cm,0cm)
      arc[x radius = 3cm, y radius = 2.5cm, start angle = 0, end angle=\n{Q'}]
      -- cycle;

Output

Use named coordinates and not hard-coded numbers, and let TikZ/TeX do the calculations for you.

Take a look at the following code. I have used named coordinates that get labeled semi-automatically. These named coordinates are used throughout the whole TikZ picture.

Hopefully, the comments in the code are helpful enough.

Code

\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings,calc,fadings,intersections}
\tikzset{
    mark at/.style args={#1 with #2}{
        decoration={
            markings,
            mark connection node=n#2,
            mark=at position #1 with {\node[font=\footnotesize\strut] (n#2) {$#2$};}
        },
        decorate
    }
}

\newcommand*{\myXradius}{3cm}
\newcommand*{\myYradius}{2.5cm}
\newcommand*{\myOverlap}{0.25cm}
\begin{document}
\begin{tikzpicture}[every node/.append style={font=\small}]
% Step 1: Place coordinates and label them.
\foreach \p/\x/\y/\pos in {%
    O/          0/          0/below left,
    A/-\myXradius/          0/below left,
    B/          0/ \myYradius/above left,
    D/          0/-\myYradius/below left,
    P/ \myXradius/          0/below right%
}   \coordinate[label={[name=n\p]\pos:$\p$}] (\p) at (\x,\y);

% Step 1': Place special coordinates and label them (same syntax as before)
\coordinate                      [label={[name=nQ]above right:$Q$}] (Q) at (30:\myXradius);
\path let \p{Q}=(Q) in coordinate[label={[name=nV]below      :$V$}] (V) at (\x{Q},0); % V is Q projected on the x axis

% Step 2: Draw the lines
\draw[mark at=1.625cm with a] (O) -- (180:\myXradius + \myOverlap);
\draw[mark at=1.625cm with b] (O) -- (  0:\myXradius + \myOverlap);
\draw                         (90:\myXradius + \myOverlap) -- (270:\myXradius + \myOverlap);
\draw[mark at=.5 with a]      (O) -- (Q);
\draw[name path=line VQ]      (V) -- (Q);

% Step 3: Draw the ellipses
\draw[                    red] (O) circle  [  radius = \myXradius                       ];
\draw[name path=ellipse, blue] (O) ellipse [x radius = \myXradius, y radius = \myYradius];

% Step 4: Find the intersection of the ellipse and the line between V and Q ...
\path[name intersections={of=line VQ and ellipse, name=Q'}]
    let \p{Q'} = (Q'-1),
        \n{Q'} = {atan2(\x{Q'}/\myXradius,\y{Q'}/\myYradius)}
    in [
        draw=green,
        fill=green!50
    ]
% ... and draw/fill it ...
    (V) -- (P)
    arc[x radius = \myXradius, y radius = \myYradius, start angle = 0, end angle=\n{Q'}]
    -- cycle
% ... and define a special name to it, as (Q'-1) will be lost after the path.
    coordinate (Q') at (Q'-1);

% Step 5: Draw little circles where the coordinates are.
\foreach \p in {O,A,B,D,P,Q,V,Q'} \fill (\p) circle [radius=1.5pt];

% Step X: I have not changes this.
\filldraw[inner color = pink!70!, outer color = blue!40!black] (0,0) -- (0.875,0) arc (0:30:0.875cm);
\node[scale = .8, fill = white] at (0.875,0) {$ae$};
\node[scale = .8] at (1,0.28) {$E$};
\end{tikzpicture}
\end{document}

Output