椭球法图示

Question

我对以下内容对应的 Latex 绘图感兴趣：

这是一张描述椭圆体方法（一种凸优化算法）背后思想的图。它是图 8.1线性最优化简介作者为 Bertsimas 和 Tsitsiklis。

我推测这可以用 tikz 或类似的东西来绘制，但我不知道如何做，所以我将不胜感激任何帮助。

编辑：感谢您的反馈。规格如下：

• $\mathbf x_t$ 位于 $E_t$ 的中心
• $\mathbf x_{t + 1}$ 位于 $_E{t + 1}$ 的中心
• 两条线平行
• $E_t$ 的右侧部分已加阴影
• $E_{t + 1}$ 是包含 $E_t$ 阴影部分的最小椭圆体
• $P$ 的确切性质并不重要。我只希望这幅图看起来与书中的图基本相同。

感恩

这是我的最小示例。遗憾的是，尺寸不太正确。

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{calc, arrows}
% \def\myellipse{(5,1) ellipse (4cm and 2cm)}

\begin{document}

\begin{figure}
    \centering
    % \includegraphics{}
    \begin{tikzpicture}[
    dot/.style = {outer sep = +0pt, inner sep = +0pt, shape = circle, draw = black, label = {#1}},
    % small dot/.style = {minimum size = 1pt, dot = {#1}},
    big dot/.style = {minimum size = 2pt, dot = {#1}},
    % line join = round, line cap = round, >=triangle 45
    ]
        % \tikz \draw \myellipse;
        \draw (0, 0) ellipse (5.5cm and 2cm);
        % \node at ($(2, 0)+(75:2 and 1)$) {$E_t$};
        \node at (-4, 0.75) {$E_t$};
        % \node at (5, 1) {$\mathbf x_t};
        % \node at (5, 1) {\textbullet};
        \node[ fill = black, big dot = {above right: \(\mathbf x_t\)}] () at (0, 0) {};
        \draw (0,0) -- (-2, 4) -- (2, -4) node[right] {$\mathbf a' \mathbf x > \mathbf a' \mathbf x_t$};
        \node[ fill = black, big dot = {above right: \(\mathbf x_{t + 1}\)}] () at (1.5, 0.15) {};
        \draw (1.5, 0.15) -- (0.75, 1.65) -- (2.25, -1.35) node[above right] {$\mathbf a' \mathbf x > b $};
    \end{tikzpicture}
    \caption{Caption}
    \label{fig:enter-label}
\end{figure}

\end{document}

Answer 1

这是使用的东西

• 图书馆

  • bbox对于有角度的椭圆/圆弧的精确边界框，
  • calc对于坐标计算，
  • intersections用于找到斜线和法向椭圆之间的交点，以及
  • ext.paths.arcto用于绘制弧线到一个点

  也

• PGFMath 函数x_radius，以及y_radius更简单的界面来访问存储在键中的值，

• PGFMath 函数返回六边形角的随机角度，rangle(i)

• 找到位于成角度的线上的点的宏\tikzonline p:a:xy = v;A通过点页和坐标=五，

• 一些假设和

• 希望在评论中解释一些其他内容。

此代码已参数化，因此您只需更改 TikZ 图片选项中的值（角度、主椭圆的半径）。但是，由于我懒得做数学运算，所以有角度的椭圆的半径是硬编码的——难道只有一个解决方案吗？

代码

\documentclass[tikz]{standalone}
\usetikzlibrary{bbox, calc, intersections, ext.paths.arcto}
\pgfmathdeclarefunction{x_radius}{0}{\pgfmathparse{\pgfkeysvalueof{/tikz/x radius}}}
\pgfmathdeclarefunction{y_radius}{0}{\pgfmathparse{\pgfkeysvalueof{/tikz/y radius}}}
\def\tikzonline#1:#2:#3=#4;{%
  intersection of {#1}--{[shift=({#2}:1)]#1} and
  \if x#3#4,0--#4,1\else0,#4--1,#4\fi}
\begin{document}
\begin{tikzpicture}[
  x radius=4, y radius=2, every label/.append style={inner sep=+.15em},
  declare function={
    ang=110;
    rangle(\x)=\x*60+rand*5;}, % randomized hexagon angles
]
\coordinate (xt) at (.5, 0) % somewhere
  % this finds the points on the line angled 120° through xt
  % that has y = ± 1.5 × y radius
 coordinate (xt-start) at (\tikzonline xt:ang:y=+1.5*y_radius;)
 coordinate (xt-end)   at (\tikzonline xt:ang:y=-1.5*y_radius;)
;

% Let's draw the normal ellipse with the given radii
% the path picture is used to fill everything right of the angled line
% but inside the ellipse.
\draw (xt) circle[] [name path=Et][
 path picture={
   \fill[lightgray] (xt-start) -| (path picture bounding box.east)
                               |- (xt-end) -- cycle;}];
% draw the angled line
\draw (xt-start) -- (xt-end)
  node[right] {$\mathbf{a'x} > \mathbf{a'x}_t$}
  [name path=xt];

% find the intersection points between ellipse and angled line
\tikzset{name intersections={of=Et and xt}}
% the intersections are known but the ellipse through those points is not
% guesstimating some radii
% use a precise bezier bounding box for the angled ellipse
\draw[rotate=ang, x radius=3, y radius=2.34, bezier bounding box]
  (intersection-1) arc to[small] coordinate (xt+1-) (intersection-2)
                   arc to[large] coordinate (xt+1+) cycle;

% since the ellipse is angled like the line
% the center of the ellipse is the center
% between the halfway points on those arcs
\draw 
 coordinate (xt+1) at ($(xt+1-)!.5!(xt+1+)$)
% let's place the other line halfway between the centers
 coordinate (2nd) at ($(xt)!.5!(xt+1)$)
    (\tikzonline 2nd:ang:y=+.7*y_radius;) coordinate (2nd-start)
 -- (\tikzonline 2nd:ang:y=-.7*y_radius;) coordinate (2nd-end)
 node [above right] {$\mathbf{a'x} > b$}
 % do this in an angled canvas for easier math
 [rotate=ang]
 % these intersections are randomized corners of the hexagon
 % but placed on the parallel line
    (intersection of 2nd-start--2nd-end and xt+1--{[shift=(rangle 1:1)]xt+1})
 -- plot[samples at = {0, 5, 4, 3}] ([shift={(rangle \x:1)}]xt+1)
 -- (intersection of 2nd-start--2nd-end and xt+1--{[shift=(rangle 2:1)]xt+1})
;

% label and draw the points
\node foreach \c/\l in {xt/"$\mathbf x_t$" above right,
                        xt+1/"$\mathbf x_{t+1}$" below}
  [shape=circle, fill, inner sep=+0pt, minimum size=+2.5pt,
   style/.expand once=\l] at (\c) {};
\end{tikzpicture}
\end{document}

输出