英文:
Shade the intersection area between 2 curves in Latex
问题
我需要绘制一个极坐标图，并填充蓝线（θ = π/8）和红色曲线（r = 4 * sqrt(2) * cos(2*θ)）之间的区域。
[图例](https://i.stack.imgur.com/TxjJz.png)
我的当前Latex代码如下（尚未填充区域）：
```latex
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=10]
\begin{scriptsize}
\draw[->](0,0)--(0.5,0) node[right] {初始线};
\draw (0.03,0.2) node[right] {$r=4\sqrt{2}\cos{2\theta}$};
\draw (0,0.01) node[above] {$O$};
\draw[blue] (0,0)--(0.35,0.15) node[right] {$\theta=\frac{\pi}{8}$};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius= {4*sqrt(2)*cos(2*\x r)});
\end{scriptsize}
\end{tikzpicture}
\end{document}
我想问如何填充区域，我搜索过但找不到解决方法（我是初学者）。谢谢您提前帮助！
我尝试通过添加以下行来绘制一个与所需位置不完全匹配的区域：
\draw[fill=gray](0.01,0)to[out=215](0.13,0.053)--(0.06,0.028)--cycle;
<details>
<summary>英文:</summary>
I'm having to draw a polar graph and shade the area between the line (theta = pi/8) (blue line) and the curve (r = 4 * sqrt(2) * cos(2*theta) (red curve).
[Figure](https://i.stack.imgur.com/TxjJz.png)
My current Latex (the area hasn't been shaded)
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=10]
\begin{scriptsize}
\draw[->](0,0)--(0.5,0) node[right] {initial line};
\draw (0.03,0.2) node[right] {$r=4\sqrt{2}\cos{2\theta}$};
\draw (0,0.01) node[above] {$O$};
\draw[blue] (0,0)--(0.35,0.15) node[right] {$\theta=\frac{\pi}{8}$};
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius= {4*sqrt(2)*cos(2*\x r)});
\end{scriptsize}
\end{tikzpicture}
\end{document}
I want to ask how to shade the area, I searched it but could not find the solution. (I'm the beginner).
Thank you in advance!
I tried to draw an area that not totally fit in the needed position by adding this line:
`\draw[fill=gray](0.01,0)to[out=215](0.13,0.053)--(0.06,0.028)--cycle;`
</details>
# 答案1
**得分**: 2
以下是已翻译的内容：
在使用 `\clip` 时非常容易。要求提供封闭的图形，例如圆形、矩形或您通过代码绘制的图形。如果在定义的末尾添加 `cycle`，则简单的 `\draw` 也可以创建多边形。顺便说一下，`[plot]` 也可以创建封闭图形。
以下是基于 `\clip` 的解决方案。有些点必须在“剪切范围”下使用两次，首先在“剪切范围”下，然后在常规层中，因此我定义并命名了一些坐标。出于同样的原因，我创建了一个宏，它被使用两次。
```latex
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{patterns.meta}
\newcommand\eq{80*sqrt(2)*cos(2*\x r)}
\begin{document}
\begin{tikzpicture}
\path
coordinate (E) at (0.6,4)
coordinate (F) at (7,3)
coordinate (P) at (3,3)
coordinate (I) at (10,0)
coordinate (O) at (0,0);
\begin{scope}
\clip[domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\draw[pattern={Lines[angle=135,distance={0.5mm}]}, pattern color=gray!50] (O)--(F)--(P)--cycle; % <--- need cycle here
\end{scope}
\node at (O) [above=1mm] {$O$}; % nodes also accept distance for labels
\node at (E) [right] {$r=4\sqrt{2}\cos{2\theta}$};
\draw[blue] (O)--(F) node[right] {$\theta=\frac{\pi}{8}$};
\draw[->,>={latex}] (O)--(I) node[right] {initial line};
\draw[red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\end{tikzpicture}
\end{document}
仅供演示，以下是如何通过您的图形剪切一个圆的示例：
\documentclass[tikz,border=2mm]{standalone}
\newcommand\eq{80*sqrt(2)*cos(2*\x r)}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip[domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\draw[fill=green] (0,0) circle (3);
\end{scope}
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\end{tikzpicture}
\end{document}
英文:
It's very easy with \clip. The requirement is to provide closed figures, such as circle, rectangle, or the one you draw by the code. A simple \draw can also create polygons if you append cycle at the end of the definition. Incidentally, [plot] creates close figures, as well.
Below is a solution based on \clip. Some points have to be used twice, first under the "clip scope" and then in regular layer, hence I have defined and named a few coordinates. For the same reason, I created a macro that is used twiced.
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{patterns.meta}
\newcommand\eq{80*sqrt(2)*cos(2*\x r)}
\begin{document}
\begin{tikzpicture}
\path
coordinate (E) at (0.6,4)
coordinate (F) at (7,3)
coordinate (P) at (3,3)
coordinate (I) at (10,0)
coordinate (O) at (0,0);
\begin{scope}
\clip[domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\draw[pattern={Lines[angle=135,distance={0.5mm}]}, pattern color=gray!50] (O)--(F)--(P)--cycle; % <--- need cycle here
\end{scope}
\node at (O) [above=1mm] {$O$}; % nodes also accept distance for labels
\node at (E) [right] {$r=4\sqrt{2}\cos{2\theta}$};
\draw[blue] (O)--(F) node[right] {$\theta=\frac{\pi}{8}$};
\draw[->,>={latex}] (O)--(I) node[right] {initial line};
\draw[red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\end{tikzpicture}
\end{document}
Just for demonstration, here's how to clip a circle by your figure:
\documentclass[tikz,border=2mm]{standalone}
\newcommand\eq{80*sqrt(2)*cos(2*\x r)}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip[domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\draw[fill=green] (0,0) circle (3);
\end{scope}
\draw[color=red,domain=0:6.28,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\eq});
\end{tikzpicture}
\end{document}
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