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如何改进立体投影

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如何改进立体投影

我正在尝试按以下方式绘制立体投影：

\documentclass{article}
\usepackage{mathtools}
\usepackage{wrapfig}
\usepackage{pgf,tikz,pgfplots}
\usetikzlibrary{decorations.pathreplacing, calligraphy}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{patterns}
\usetikzlibrary{calc}

\begin{document}
    \begin{figure}
        \centering
        \begin{tikzpicture}
            \coordinate (A) at (3,-0.25);
            \coordinate (P) at (0,2);
            \draw (0,0) circle (2);
            \draw (0,0) ellipse (2 and 0.5);
            \draw (-4.5,-1) -- (3.5,-1) -- (4.5,1) node[anchor=south east] {\scriptsize$ z=0 $} -- (-3.5,1) -- cycle;
            \draw (A) -- (P) coordinate[pos=0.47](B);
            \path (A) node[circle, fill, inner sep=1pt, label=below:{\scriptsize$ P(x,y,0) $}]{};
            \path (B) node[circle, fill, inner sep=1pt, label=left:{\scriptsize$ (\xi,\eta,\zeta) $}]{};
            \path (P) node[circle, fill, inner sep=1pt, label=above:{\scriptsize$ (0,0,1) $}]{};
            \draw [dashed] (-2,0) -- (2,0);
        \end{tikzpicture}
    \end{figure}
\end{document}

我想要实现的是沿着下图中的红线生成虚线： 在此处输入图片描述

另外，我想沿着蓝线添加另一条 MWE 中没有的虚线。

答案1

您只需要在不同的路径中绘制虚线和实线部分，因此对于弧形路径使用arcs 而不是s。circle

30 度角是通过反复试验找到的，但如果有需要也可以计算出来。

在此处输入图片描述

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots} % loads tikz which loads pgf
\begin{document}
        \begin{tikzpicture}
            \coordinate (A) at (3,-0.25);
            \coordinate (P) at (0,2);

            \draw (0:2cm)   arc[radius=2cm,start angle=0,end angle=180]
                  (210:2cm) arc[radius=2cm,start angle=210,end angle=330];
            \draw (180:2cm) arc[x radius=2cm, y radius=0.5cm, start angle=180,end angle=360];

            \draw [dashed] (210:2cm) 
                  arc[start angle=210,delta angle=-30,radius=2cm]
                  arc[start angle=180,delta angle=-180,x radius=2cm,y radius=0.5cm]
                  arc[start angle=0,delta angle=-30,radius=2cm];

            \draw [dashed] (80:2cm and 0.5cm) -- (260:2cm and 0.5cm);
            \draw [dashed] (150:2cm) coordinate(ul) -- (30:2cm) coordinate(ur);

            \draw (-4.5,-1) -- (3.5,-1) -- (4.5,1) node[anchor=south east] {\scriptsize$ z=0 $} -- (ur) (ul) -- (-3.5,1) -- (-4.5,-1);

            \draw (A) -- (P) coordinate[pos=0.47](B);
            \path (A) node[circle, fill, inner sep=1pt, label=below:{\scriptsize$ P(x,y,0) $}]{};
            \path (B) node[circle, fill, inner sep=1pt, label=left:{\scriptsize$ (\xi,\eta,\zeta) $}]{};
            \path (P) node[circle, fill, inner sep=1pt, label=above:{\scriptsize$ (0,0,1) $}]{};
            \draw [dashed] (-2,0) -- (2,0);

        \end{tikzpicture}
\end{document}

答案2

这是立体投影的正交投影。即使用具有实际立体投影的 3d 坐标并通过正交投影投影到屏幕上。虚线是通过一系列剪辑和反向剪辑实现的。

\documentclass{article}
\usepackage{tikz}
% based on 
% https://tex.stackexchange.com/a/38995/121799 
% https://tex.stackexchange.com/a/76216 
% https://tex.stackexchange.com/a/59168/194703 
% https://tex.stackexchange.com/q/448920/194703 
\makeatletter 
\tikzset{ 
reuse path/.code={\pgfsyssoftpath@setcurrentpath{#1}} 
} 
\tikzset{even odd clip/.code={\pgfseteorule}, 
protect/.code={ 
\clip[overlay,even odd clip,reuse path=#1] 
(current bounding box.south west) rectangle (current bounding box.north east)
%(-16383.99999pt,-16383.99999pt) rectangle (16383.99999pt,16383.99999pt)
; 
}} 
\makeatother 
\usetikzlibrary{3d,perspective}
\begin{document}
\pgfmathsetmacro{\myaz}{10}
\begin{tikzpicture}[declare function={%
        stereox(\x,\y)=2*\x/(1+\x*\x+\y*\y);%
        stereoy(\x,\y)=2*\y/(1+\x*\x+\y*\y);%
        stereoz(\x,\y)=(-1+\x*\x+\y*\y)/(1+\x*\x+\y*\y);},scale=2.5,
        line join=round,line cap=round,
        dot/.style={circle,fill,inner sep=1pt}]
 \path[save path=\pathSphere] (0,0) circle[radius=1];
 \begin{scope}[3d view={\myaz}{15}]
  \draw (-2,2) -- (-2,-2) coordinate (bl) -- (2,-2) coordinate (br)-- (2,2)
  node[above left]{$z=0$};
  \begin{scope}
   \tikzset{protect=\pathSphere}
   \draw (-2,2) -- (2,2);
  \end{scope}
  \begin{scope}
   \clip[reuse path=\pathSphere];
   \draw[dashed] (-2,2) -- (2,2);
  \end{scope}
  \begin{scope}[canvas is xy plane at z=0]
   \draw[dashed] (\myaz:1) arc[start angle=\myaz,end angle=\myaz+180,radius=1];
   \draw (\myaz:1) arc[start angle=\myaz,end angle=\myaz-180,radius=1];
   \path[save path=\pathPlane] (\myaz:2) -- (\myaz+180:2) --(bl) -- (br) -- cycle;
   \begin{scope}
    \clip[use path=\pathPlane];
    \draw[dashed,use path=\pathSphere];
   \end{scope}
   \begin{scope}
    \tikzset{protect=\pathPlane}
    \draw[use path=\pathSphere];
   \end{scope}
  \end{scope}
  \draw (1.5,-1,0) node[dot,label=below:{$P(x,y,0)$}]{}
  -- ({stereox(1.5,-1)},{stereoy(1.5,-1)},{stereoz(1.5,-1)})
   node[dot,label=below left:{$(\xi,\eta,\zeta)$}](I){};
  \draw[dashed] (I) -- (0,0,1) node[dot,label=above:{$(0,0,1)$}]{}; 
 \end{scope}
\end{tikzpicture}
\end{document}

在此处输入图片描述

可以改变平面上点的视图参数和坐标。（这些都包含在一个动画中，但参数可以单独改变。）

\documentclass[tikz,border=3mm]{standalone}
% based on 
% https://tex.stackexchange.com/a/38995/121799 
% https://tex.stackexchange.com/a/76216 
% https://tex.stackexchange.com/a/59168/194703 
% https://tex.stackexchange.com/q/448920/194703 
\makeatletter 
\tikzset{ 
reuse path/.code={\pgfsyssoftpath@setcurrentpath{#1}} 
} 
\tikzset{even odd clip/.code={\pgfseteorule}, 
protect/.code={ 
\clip[overlay,even odd clip,reuse path=#1] 
(current bounding box.south west) rectangle (current bounding box.north east)
; 
}} 
\makeatother 
\usetikzlibrary{3d,perspective}
\begin{document}
\foreach \X in {5,15,...,355}
{\pgfmathsetmacro{\myaz}{10+10*sin(\X)}
\begin{tikzpicture}[declare function={%
        stereox(\x,\y)=2*\x/(1+\x*\x+\y*\y);%
        stereoy(\x,\y)=2*\y/(1+\x*\x+\y*\y);%
        stereoz(\x,\y)=(-1+\x*\x+\y*\y)/(1+\x*\x+\y*\y);
        Px=1.75+0.5*sin(2*\X);Py=-1.5+0.5*cos(2*\X);amax=2.5;},scale=2.5,
        line join=round,line cap=round,
        dot/.style={circle,fill,inner sep=1pt}]
 \pgfdeclarelayer{background} 
 \pgfdeclarelayer{foreground} 
 \pgfsetlayers{background,main,foreground}
 \path[use as bounding box] (-3.5,-2) rectangle (3.5,2);
 \path[save path=\pathSphere,ball color=gray,fill opacity=0.6] 
    (0,0) circle[radius=1];
 \begin{scope}[3d view={\myaz}{15}]
  \draw (-amax,amax) -- (-amax,-amax) coordinate (bl) -- (amax,-amax) 
  coordinate (br)-- (amax,amax)
  node[above left]{$z=0$};
  \begin{scope}
   \tikzset{protect=\pathSphere}
   \draw (-amax,amax) -- (amax,amax);
  \end{scope}
  \begin{scope}
   \clip[reuse path=\pathSphere];
   \draw[dashed] (-amax,amax) -- (amax,amax);
  \end{scope}
  \begin{scope}[canvas is xy plane at z=0]
   \draw[dashed] (\myaz:1) arc[start angle=\myaz,end angle=\myaz+180,radius=1];
   \draw (\myaz:1) arc[start angle=\myaz,end angle=\myaz-180,radius=1];
   \path[save path=\pathPlane] (\myaz:amax) -- (\myaz+180:amax) --(bl) -- (br) -- cycle;
   \begin{scope}
   %\begin{pgfonlayer}{background}   
    \clip[use path=\pathPlane];
    \draw[dashed,use path=\pathSphere];
   %\end{pgfonlayer}
   \end{scope}
   \begin{scope}
    \tikzset{protect=\pathPlane}
    \draw[use path=\pathSphere];
   \end{scope}
   \begin{pgfonlayer}{background}
    \fill[blue!30,fill opacity=0.6]
     (\myaz:1) arc[start angle=\myaz,end angle=\myaz-180,radius=1]
     -- (-amax,0) -- (-amax,amax) -- (amax,amax) -- (amax,0) -- cycle;
   \end{pgfonlayer}
    \fill[blue!30,fill opacity=0.6]
     (\myaz:1) arc[start angle=\myaz,end angle=\myaz-180,radius=1]
     -- (-amax,0) -- (-amax,-amax) -- (amax,-amax) -- (amax,0) -- cycle;
  \end{scope}
  \draw (Px,Py,0) node[dot,label=below:{$P(x,y,0)$}]{}
  -- ({stereox(Px,Py)},{stereoy(Px,-1)},{stereoz(Px,Py)})
   node[dot,label=below left:{$(\xi,\eta,\zeta)$}](I){};
  \begin{pgfonlayer}{background} 
   \draw[dashed] (I) -- (0,0,1) node[dot,label=above:{$(0,0,1)$}]{};
  \end{pgfonlayer} 
 \end{scope}
\end{tikzpicture}}
\end{document}

在此处输入图片描述

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