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最大能量损失 #1

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最大能量损失 #1

为了提供进一步的说明（为您节省一些阅读时间），我们尝试以参数方式定义第二个图表，以便截断两个平面部分，只留下可用于图表的部分。希望这有帮助！谢谢

对这个问题所采用的技术感兴趣：如何绘制两个相交的圆柱体？。

具体来说，由于我仍在学习 Asymptote，我想使用 pgfplots 方法绘制三个圆柱体的相交表面。我通过摆弄代码得到了三个相交圆柱体的图表，但在创建它们的相交表面时却遇到了困难。

这是我所拥有的相交圆柱体的图表，供参考（当然，这要归功于它的灵感。） - 请参阅底部的 MWE #1：

在此处输入图片描述

这就是我目前所得到的（如果我能解决一个部分，那么我就能解决剩下的部分） - 参见 MWE #2：

在此处输入图片描述

我想要的是这种形状的东西，除了对应的是三个圆柱体而不是两个圆柱体——归功于斯图尔特：

在此处输入图片描述

最大能量损失 #1

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=3]
  \begin{axis}[%
        axis equal,
        enlargelimits = true,
        samples = 45, samples y = 45,
        axis lines = none, ticks = none,
        cyl/.style = {%
                surf,
                black!30!,
                variable = \u,
                variable y = \v,
                z buffer = sort,
                faceted color=black!70!,
                },
        %view/h = 125, view/v = 25
        ]

    \addplot3[%         (-) Z-SEMIAXIS
        cyl,
        domain = -3:3,
        y domain = 0:360,
        ] ({cos(v)}, {sin(v)}, {min(u,abs(cos(v)),abs(sin(v)))});

    \addplot3[%         (-) X-SEMIAXIS
        cyl,
        domain = -3:3,
        y domain = 0:360,
        ] ({min(u,-abs(cos(v)),-abs(sin(v)))}, {cos(v)}, {sin(v)});

    \addplot3[%         (+) Y-SEMIAXIS
        cyl,
        domain = 0:360,
        y domain = -3:3,
        ] ({cos(u)}, {max(v,abs(cos(u)),abs(sin(u)))}, {sin(u)});

    \addplot3[%         (+) X-SEMIAXIS
        cyl,
        domain = -3:3,
        y domain = 0:360,
        ] ({max(u,abs(cos(v)),abs(sin(v)))}, {cos(v)}, {sin(v)});

    \addplot3[%         (+) X-SEMIAXIS
        cyl,
        domain = -3:3,
        y domain = 0:360,
        ] ({cos(v)}, {sin(v)}, {max(u,abs(cos(v)),abs(sin(v)))});

    \addplot3[%         (-) Y-SEMIAXIS
        cyl,
        domain = 0:360,
        y domain = -3:3,
        ] ({cos(u)}, {min(-abs(cos(u)),-abs(sin(u)),v)}, {sin(u)});

    \end{axis}
\end{tikzpicture}
\end{center}
\end{document}

最大能量损失 #2

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=2]
  \begin{axis}[%
        axis equal,
        enlargelimits = true,
        samples = 45, samples y = 45,
        axis lines = none, ticks = none,
        cyl/.style = {%
                surf,
                black!30!,
                variable = \u,
                variable y = \v,
                z buffer = sort,
                faceted color=black!70!,
                },
        %view/h = 125, view/v = 25
        ]

    \addplot3[%         (-) Z-SEMIAXIS
        cyl,
        domain = -3:3,
        y domain = 0:360,
      restrict z to domain=-2:2,
        ] ({max(abs(u),cos(v))}, {sin(v)}, {u});

    \end{axis}
\end{tikzpicture}
\end{center}
\end{document}

由@hpekristiansen 编辑：

这是三个圆柱相交的图片 三个圆柱体的交点

在此处找到：https://abel.math.harvard.edu/archive/21a_summer_06/handouts/3cylinder.pdf

答案1

\documentclass[tikz, border=1cm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\begin{document}
\begin{tikzpicture}[line join=round]
\begin{axis}[
view={20}{40},
axis equal,
axis lines=none,
colormap/blackwhite,
cylinder/.style={
  surf, 
  domain=-1:1,
  samples=41,
  variable=\u, variable y=\v,
  z buffer=sort,
  ultra thin,
},
]

\addplot3[
cylinder, red!40,
y domain=0:270, %adjust domain to show front petals dependent on `view`
samples y=61,
x filter/.expression={
  (u>0)?
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(min( sqrt(1-cos(v)^2), sqrt(1-sin(v)^2)))):
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(max(-sqrt(1-cos(v)^2),-sqrt(1-sin(v)^2))))
},
] (u, {cos(v)}, {sin(v)});

\addplot3[
cylinder, green!40,
y domain=-90:90, %adjust domain to show front petals dependent on `view`
samples y=41,
y filter/.expression={
  (u>0)?
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(min( sqrt(1-cos(v)^2), sqrt(1-sin(v)^2)))):
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(max(-sqrt(1-cos(v)^2),-sqrt(1-sin(v)^2))))
},
] ({sin(v)}, u, {cos(v)});

\addplot3[
cylinder, blue!40,
y domain=180:360, %adjust domain to show front petals dependent on `view`
samples y=41,
z filter/.expression={
  (u>0)?
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(min( sqrt(1-cos(v)^2), sqrt(1-sin(v)^2)))):
  ((cos(v)^2+u^2<1)&&(sin(v)^2+u^2<1)?u:(max(-sqrt(1-cos(v)^2),-sqrt(1-sin(v)^2))))
},
] ({cos(v)}, {sin(v)}, u);

\end{axis}
\end{tikzpicture}
\end{document}

三个圆柱体的红绿蓝相交点

答案2

在此处输入图片描述

一种可能性是分别构建每个“半面”，并根据视点逐层添加。面的参数化形式为 (cos(u)、sin(u)、max(cos(u)、abs(sin(u))、v))，其中 -90<=u<=90 且 0<=v<=1.1（对于与 Oz 轴平行的圆柱体）。

将循环排列坐标以获得另外两个圆柱体。

评论。 顺序很重要，我只知道如何从观察者的角度考虑并检查输出来手工完成。

我在两个圆柱的交点下方插入；在这种情况下，第三个坐标更简单——mxax(cos(u), v)——因为只有两个相交的表面。但它表明三个相交圆柱的情况是一种自然的概括。

在此处输入图片描述

代码

\documentclass[11pt, margin=10pt]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{colorbrewer}

\pgfplotsset{compat=1.17}
\usetikzlibrary[pgfplots.colormaps]
\begin{document}

\pgfplotsset{%
  cylinder piece/.style args={u domain=#1, cmap=#2}{%
    opacity = 1,
    surf,
    colormap/#2,
    shader=interp,
    z buffer = sort,
    samples = 33,  % 1 mod 4
    variable = \u,
    variable y = \v,
    domain = #1,
    y domain = 0:1.1,
  }
}
\iffalse  % 2 intersectiong cylinders
\begin{tikzpicture}
  \begin{axis}[axis equal,
    axis lines=none,
    view={130}{28}]

    % z cylinder x<0
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({-cos(u)}, {sin(u)}, {-min(cos(u), v)});
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({-cos(u)}, {sin(u)}, {min(cos(u), v)});

    % x cylinder z<0
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({-min(sin(u), v)}, {cos(u)}, {-sin(u)});
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({min(sin(u), v)}, {cos(u)}, {-sin(u)});
    
    % x cylinder z>0
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({-min(sin(u), v)}, {cos(u)}, {sin(u)});
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({min(sin(u), v)}, {cos(u)}, {sin(u)});
    
    % z cylinder x>0
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {-min(cos(u), v)});
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {min(cos(u), v)});
  \end{axis}
\end{tikzpicture}
\fi


% \iffalse  % 3 intersecting cylinders
\begin{tikzpicture}
  \begin{axis}[axis equal,
    axis lines=none,
    view={130}{20}]

    % y cylinder, x < 0
    \addplot3[cylinder piece={u domain={-90:90}, cmap=winter}]
    ({-cos(u)}, {-min(min(cos(u), abs(sin(u))), v)}, {sin(u)});
    \addplot3[cylinder piece={u domain={-90:90}, cmap=winter}]
    ({-cos(u)}, {min(min(cos(u), abs(sin(u))), v)}, {sin(u)});

    % z cylinder, x < 0
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({-cos(u)}, {sin(u)}, {-min(min(cos(u), abs(sin(u))), v)});
    \addplot3[cylinder piece={u domain={-90:90}, cmap=summer}]
    ({-cos(u)}, {sin(u)}, {min(min(cos(u), abs(sin(u))), v)});

    % x cylinder z < 0
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({-min(sin(u), abs(cos(u)), v)}, {cos(u)}, {-sin(u)});
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({min(sin(u), abs(cos(u)), v)}, {cos(u)}, {-sin(u)});

    % x cylinder z > 0
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({-min(sin(u), abs(cos(u)), v)}, {cos(u)}, {sin(u)});
    \addplot3[cylinder piece={u domain={0:180}, cmap=autumn}]
    ({min(sin(u), abs(cos(u)), v)}, {cos(u)}, {sin(u)});
    
    % z cylinder, x > 0
    \addplot3[cylinder piece={u domain={-90:0}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {-min(min(cos(u), abs(sin(u))), v)});
    \addplot3[cylinder piece={u domain={-90:0}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {min(min(cos(u), abs(sin(u))), v)});

    % y cylinder, x > 0
    \addplot3[cylinder piece={u domain={-90:90}, cmap=winter}]
    ({cos(u)}, {-min(min(cos(u), abs(sin(u))), v)}, {sin(u)});
    \addplot3[cylinder piece={u domain={-90:90}, cmap=winter}]
    ({cos(u)}, {min(min(cos(u), abs(sin(u))), v)}, {sin(u)});
    
    % half z cylinder, x > 0
    \addplot3[cylinder piece={u domain={0:90}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {-min(min(cos(u), abs(sin(u))), v)});
    \addplot3[cylinder piece={u domain={0:90}, cmap=summer}]
    ({cos(u)}, {sin(u)}, {min(min(cos(u), abs(sin(u))), v)});
  \end{axis}
\end{tikzpicture}
% \fi

\end{document}

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