如何精确的绘制二维向量的两个分向量?

发布于 2024-03-30 19:33:24

TIKZ能自动识别、并绘制曲线上某点的切线、法向量。那么图中P点的法向量,如何将法向量vector分解到xy坐标系下精确绘制出红色(i,j)子向量。现在图中仅凭肉眼观察“凑合”的粗略画法。
quest800.png

\documentclass{standalone}
\usepackage{xcolor}
\usepackage{tikz}
\usetikzlibrary{decorations.markings} 
\begin{document}
    \begin{tikzpicture}[
        tangent/.style = {    % 定义tangent样式,参数#1是切点在曲线上的相对位置
            decoration = {    % 对path进行装饰(decorate)
                markings,    % 启用marking
                mark = at position #1    % 在指定位置#1添加marking
                with{    % marking的具体内容
                    % 定义(0, 0)为tangent point-xxx,即切点
                    \coordinate (tangent point-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0, 0);
                    % 定义(1, 0)为tangent unit vector-xxx,即单位切向量
                    \coordinate (tangent unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1, 0);
                    % 定义(0, 1)为tangent orthogonal unit vector-xxx,即单位法向量
                    \coordinate (tangent orthogonal unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0, 1);
                }
        },
postaction = decorate   % 设置postaction,从而保留path
},
use tangent/.style = {    % 定义use tangent样式,参数#1是切点的序号
shift = (tangent point-#1),     % 以tangent point-xxx为原点
x = (tangent unit vector-#1),    % 以tangent unit vector-xxx为x轴单位向量
y = (tangent orthogonal unit vector-#1)    % 以tangent orthogonal unit vector-xxx为y轴单位向量
},use tangent/.default = 1]
\draw[dotted,black!10] (0,1.55) -- (2, 1.55); %辅助线,随时删除
\draw[dotted,black!10] (0.9,0) -- (0.9,1.7); %辅助线,随时删除
\draw[->] (-1,0) -- (3,0) node[anchor=south east]{$x$};
\draw[->] (0,-1) -- (0,2) node[anchor=south west]{$y$};    
\draw[tangent = 0.65] (0, 0) .. controls (1, 0) and (1, 1) .. (2, 0.5);
\fill[use tangent] circle(1pt) node[below=1] {$p$};
\draw[,black!20,use tangent] (-1.5, 0) -- (1.5, 0);%切线
\draw[->, use tangent] (0, 0) -- (0, 1)node[scale=0.4,above=2] {$vector$};; %法线
\draw[red,->, use tangent] (0, 0) -- (0.31,0.9) node[right=1] {$j$}; 
\draw[red,->, use tangent] (0, 0) -- (-0.325,0.125) node[left=1] {$i$}; 
\end{tikzpicture}
\end{document}            

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1 个回答
雾月
雾月 2024-03-30
这家伙很懒,什么也没写!

可以保存切点和向量终点的坐标,然后利用 perpendicular 坐标系统即可画出。
ans-145-tikz.jpeg

\documentclass[tikz,border=5pt]{standalone}
\usetikzlibrary{decorations.markings}

\makeatletter 
\def\my@tangentparse#1{\in@{ at}{#1}%
  \ifin@\my@tangentparse@#1\@nil\else\my@tangentparse@ last at#1\@nil\fi}
\def\my@tangentparse@#1 at#2\@nil{\pgfkeysalso{postaction=decorate,
  decoration = {    % 对path进行装饰(decorate)
    markings,    % 启用marking
    mark = at position #2    % 在指定位置#1添加marking
      with{    % marking的具体内容
        % 定义(0, 0)为tangent point xxx,即切点
        \coordinate (tangent point #1) at (0, 0);
        % 定义(1, 0)为tangent unit vector xxx,即单位切向量
        \coordinate (tangent unit vector #1) at (1, 0);
        % 定义(0, 1)为tangent orthogonal unit vector xxx,即单位法向量
        \coordinate (tangent orthogonal unit vector #1) at (0, 1);
      }
  },
}}
\tikzset{
  tangent/.code = \my@tangentparse{#1},
  use tangent/.style = {    % 定义use tangent样式,参数#1是切点的序号
    shift = (tangent point #1),     % 以tangent point-xxx为原点
    x = (tangent unit vector #1),    % 以tangent unit vector-xxx为x轴单位向量
    y = (tangent orthogonal unit vector #1)% 以tangent orthogonal unit vector-xxx为y轴单位向量
  },
  use tangent/.default = last,
}
\makeatother
\begin{document}
\begin{tikzpicture}
\draw[->] (-1,0) -- (3,0) node[anchor=south east]{$x$};
\draw[->] (0,-1) -- (0,2) node[anchor=south west]{$y$};
\draw[tangent=A at 0.65,tangent=B at 0.8] (0, 0) .. controls (1, 0) and (1, 1) .. (2, 0.5);
\fill[use tangent=A] circle(1pt) node[below=1] {$p$};
\draw[black!20,use tangent=A] (-1.5, 0) -- (1.5, 0);%切线
\draw[->, use tangent=A] (0, 0) -- (0, 1) coordinate(L) node[scale=0.4,above=2] {$vector$}; %法线

\draw[dotted,black!10, shorten <=-1cm] (L) -- (0,0 |- L); %辅助线
\draw[dotted,black!10] (L) -- (0,0 -| L); %辅助线

\draw[red,->] (tangent point A) -- (L |- tangent point A);
\draw[red,->] (tangent point A) -- (L -| tangent point A);

\draw[->, use tangent=B] (0, 0) -- (0, 1) coordinate(L);
\draw[red,->] (tangent point B) -- (L |- tangent point B);
\draw[red,->] (tangent point B) -- (L -| tangent point B);

\end{tikzpicture}
\end{document}

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