Ellipse with eigenvectors and eigenvalus in tikz

Question

I'm trying to make ellipse with eigenvectors and eigenvalues in tikz. I'm having tough time with managing my braces. I wonder if there is any analyticall approach to adjust braces. Thanks

\documentclass{standalone}
\usepackage{pgfplots}
\usetikzlibrary{shapes.geometric}
\pgfplotsset{compat=1.7}

\begin{document}

\begin{tikzpicture}
\draw[rotate=45, line width = 0.5mm] (0, 0) ellipse (4.5cm and 2.0cm);

\draw[line width = 0.5mm, dashed, ->] (-5, 0) -- (5, 0) node[right]{$X_{1}$};
\draw[line width = 0.5mm, dashed, ->] (0, -5) -- (0, 5) node[above]{$X_{2}$};
\draw[line width = 0.5mm, ->] (-4, -4) -- (4, 4) node[above right]{$e_1$};
\draw[line width = 0.5mm, ->] (3, -3) -- (-4, 4) node[above left]{$e_2$};
\draw [decorate, decoration = {brace, amplitude = 15pt, mirror, raise =4pt}, yshift = 0pt]
(3, 3.2) -- (0, 0) node [black, midway, xshift = -0.5cm, yshift = 0.8cm ] {$\frac{c}{\sqrt{\lambda_1}}$};

\draw [decorate, decoration = {brace, amplitude = 15pt, mirror, raise =4pt}, yshift = 0pt]
(-1.4, 1.4) -- (0, 0) node [black, midway, xshift = -0.9cm, yshift = -0.3cm ] {$\frac{c}{\sqrt{\lambda_2}}$};

\end{tikzpicture}

\end{document}

Answer 1

You have loaded the pgfplots package, although you did not use it. I have removed it and have loaded only TikZ instead (in this case by the class option tikz).

Use styles!

I have introduced a few. These are:

• elli=<angle>:<x radius> and <y radius>

  The elli style results in a node in the shape of an ellipse with the assigned radii rotated by <angle>.

  The style also uses outer sep=0pt to move the anchors to the middle of the line.

• elli node=<angle>

  The elli node style should be used for the nodes for the braces (the eigenvalues?). The not-drawn shape circle is used to place the node more precisely. This is done in two ways:

  1. The anchor is set to <angle>-90, so that the node is placed rectangular to the decorated line.

  2. The node is shifted in the opposite direction, <angle>+90 by the amount of the amplitude and the raise amount (the raise amount is not stored in a macro like the amplitude amount so I stored in manually with the addition to /pgf/decoration/raise).

  Take a look at this animation to see how you can overwrite the anchor to replace the node (I used this for the y eigenvalue). It may also needed to use the shift options of TikZ anyway.

  

• eigen={<amplitude>}{<raise>}

  This style does all bracing with the given values for the amplitude and the raise option. The default values are 15pt and 4pt (as in your example).

• The styles axis and normal axis should be self-explanatory.

Notes (Further Enhancements)

• Rename the styles if you have more fitting names, say eigenvalue instead of elli node.

• I noticed now that the mirror option is not taken into account for the placement of the elli node. Meaning that the nodes are always placed on the left side. This could be fixed quite easily.

• I have used the ellipsoid node’s anchors to draw the lines e₁ and e₂. I am not entirely sure how long the lines should be. This is just one option of many to place lines relative to another object.

  The normal axes are not touched (except for the style).

Code

\documentclass[tikz]{standalone}
\usetikzlibrary{
    shapes.geometric,
    decorations.pathreplacing
}
\tikzset{
    elli/.style args={#1:#2and#3}{
        draw,
        shape=ellipse,
        rotate=#1,
        minimum width=2*#2,
        minimum height=2*#3,
        outer sep=0pt,
    },
    /pgf/decoration/raise/.append code={
        \def\tikzdecorationsbrace{#1}
    },
    elli node/.style={
        circle,
        black,
        draw=none,
        midway,
        anchor=#1-90,
        inner sep=0pt,
        shift=(#1+90:\tikzdecorationsbrace+\pgfdecorationsegmentamplitude)
    },
    eigen/.style 2 args={
        decorate,
        decoration={
            brace,
            amplitude=#1,
            mirror,
            raise=#2,
        },
    },
    eigen/.default={15pt}{4pt},
    axis/.style={
        line width=.5mm,
        ->,
    },
    normal axis/.style={
        axis,
        dashed,
    }
}
\begin{document}

\begin{tikzpicture}
\node[elli=45:4.5cm and 2.0cm, line width = 0.5mm] at (0, 0) (e) {};

\draw[normal axis] (-5, 0) -- (5, 0) node[right] {$X_{1}$};
\draw[normal axis] (0, -5) -- (0, 5) node[above] {$X_{2}$};

\draw[axis] ([shift={(45:-5.5cm)}]    e.center) -- ([shift={(45:5.5cm)}]    e.center) node[above right] {$e_1$};
\draw[axis] ([shift={(90+45:-5.5cm)}] e.center) -- ([shift={(90+45:5.5cm)}] e.center) node[above left]  {$e_2$};


\draw[red,eigen] (e.east) -- (e.center) 
    node[elli node=45] {$\frac{c}{\sqrt{\lambda_1}}$};

\draw[red,eigen] (e.north) -- (e.center) 
    node[elli node=45+90, anchor=-15] {$\frac{c}{\sqrt{\lambda_2}}$};
\end{tikzpicture}
\end{document}

Output

Answer 2

You should use scope environment to draw all the stuff that will rotate, and place the coordinate for the non rotating text.

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.pathreplacing}

\begin{document}
\begin{tikzpicture}
\begin{scope}[remember picture,rotate=45]
    \coordinate (O) at (0,0);
    \coordinate (A) at (4.5,0);
    \coordinate (B) at (0,2);
    % draw ellipse
    \draw[line width = 0.5mm] (O) ellipse (4.5cm and 2.0cm);
    % draw axis
    \draw[line width = 0.5mm, ->] (-5, 0) -- (5, 0) coordinate[right] (e1);
    \draw[line width = 0.5mm, ->] (0, -5) -- (0, 5) coordinate[above] (e2);
    % calculate brace points for major axis
    \path (O) -- (A) coordinate[pos=.02] (a1) coordinate[pos=.98] (a2);
    \draw[decorate, decoration = {brace, amplitude = 15pt, mirror, raise =4pt}, yshift = 0pt]
        (a2) -- (a1) coordinate[midway] (tl1);
    \coordinate (l1) at ([yshift=1cm]tl1);
    % calculate brace points for minor axis
    \path (O) -- (B) coordinate[pos=.05] (b1) coordinate[pos=.95] (b2);
    \draw[decorate, decoration = {brace, amplitude = 15pt, mirror, raise =4pt}, yshift = 0pt]
        (b2) -- (b1) coordinate[midway] (tl2);
    \coordinate (l2) at ([xshift=-.9cm]tl2);
\end{scope}
\draw[line width = 0.5mm, dashed, ->] (-5, 0) -- (5, 0) node[right]{$X_{1}$};
\draw[line width = 0.5mm, dashed, ->] (0, -5) -- (0, 5) node[above]{$X_{2}$};
\node[anchor=south west] at (e1) {$e_1$};
\node[anchor=south east] at (e2) {$e_2$};
\node[anchor=south west] at (l1) {$\frac{c}{\sqrt{\lambda_1}}$};
\node[anchor=south east] at ([xshift=.1cm]l2) {$\frac{c}{\sqrt{\lambda_2}}$};   
\end{tikzpicture}
\end{document}

Edit: modifications as suggested

Answer 3

\documentclass[pstricks,border=3pt]{standalone}
\usepackage{pst-plot,pst-node,amsmath}

\pstVerb
{
    /PtCm {72.27 div 2.54 mul} def
    /Radius 11 def
}
\psset
{
    dimen=middle,
    nodesep=2pt,
    linearc=5pt,
    arm=4pt,
}

\begin{document}
\begin{pspicture}(-3.75,-3.75)(4,4)
\psaxes[labels=none,ticks=none,arrowinset=0,linestyle=dashed]{->}(0,0)(-3.75,-3.75)(3.25,3.5)[$X_1$,0][$X_2$,90]
\rput{45}(0,0){
        \psaxes[labels=none,ticks=none,arrowinset=0]{->}(0,0)(-3.5,-3.5)(3.5,3.5)
            [{\rput{*0}(5pt,0){$e_1$}},0]
            [{\rput{*0}(0,5pt){$e_2$}},90]
        \psellipse[dimen=middle](0,0)(3,1.5)
        \psset{linecolor=gray,angleA=90,angleB=270}
        \pcdiag(0,0)(1.5,12pt)
        \pcdiag(3,0)(1.5,12pt)
        \rput{*0}(!Radius PtCm 100 PtoC exch 1.5 add exch 12 PtCm add){$\tfrac{c}{\sqrt{\lambda_1}}$}
        \psset{angleA=180,angleB=0}
        \pcdiag(0,1.5)(-12pt,0.75)
        \pcdiag(0,0)(-12pt,0.75)
        \rput*{*0}(!Radius PtCm 200 PtoC exch 12 PtCm sub exch 0.75 add){$\tfrac{c}{\sqrt{\lambda_2}}$}
}
\end{pspicture}
\end{document}