如何消除Z方向的子向量绘制误差？

谢谢老师的完美指导。

为了突出已知点上法向量的空间构造，我又补了几条虚线，但是都靠肉眼绘制，如果已知点坐标变化了，就不能实现了。老师您有空在帮我看看，谢谢。

\documentclass[svgnames]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \makeatletter
    \newcommand\computegrad[4][0.00025]{% [delta], function, x, y
        \begingroup\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
        \def\@tempdelta{#1}%
        \pgfmathparse{#2(#3,#4)}\let\@tempz\pgfmathresult
        \pgfmathparse{#2(#3+\@tempdelta,#4)}\let\@temppxa\pgfmathresult
        \pgfmathparse{#2(#3-\@tempdelta,#4)}\let\@temppxb\pgfmathresult
        \edef\@temppx{\fpeval{(\@temppxa-\@temppxb)/(1*\@tempdelta)}}%
        \pgfmathparse{#2(#3,#4+\@tempdelta)}\let\@temppya\pgfmathresult
        \pgfmathparse{#2(#3,#4-\@tempdelta)}\let\@temppyb\pgfmathresult
        \edef\@temppy{\fpeval{(\@temppya-\@temppyb)/(1*\@tempdelta)}}%
        \edef\@tempu{\fpeval{(\@temppx,\@temppy,-1)/sqrt(\@temppx^2+\@temppy^2+1)}}%
        \edef\?{\endgroup\def\noexpand\pgfmathresult{\@tempu}%
            \def\noexpand\pgfmathresulta{\fpeval{1*(#3,#4,\@tempz)-\@tempu}}}%
        \?}
    \def\gradcomponent#1#2#3{%
        \edef\pgfmathresult{\fpeval{#2+\expanded{\noexpand\gradcomponent@#1}{#3}}}}
    \def\gradcomponent@(#1,#2,#3)#4{\ifcase                                                
        \if#4x 1 \else\if#4y 2 \else\if#4z 3 \else 4 \fi\fi\fi
        \or (0,#2,#3)\or (#1,0,#3)\or (#1,#2,0)\else(#1,#2,#3)\fi}
    %%%%%%%%%%%%%%%%%green
    \newcommand\computegraf[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}}
    %%%%%%%%%%%%%%%%%yellow
    \newcommand\computegrag[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{y}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%red
    \newcommand\computegrah[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{x}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%blue
    \newcommand\computegral[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{z}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%blue1
\newcommand\computegrat[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
    \let\pgfmathresultb\pgfmathresult
    \gradcomponent\pgfmathresult\pgfmathresulta{z}%
    \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}    
    \makeatother    
    \begin{tikzpicture}
        [declare function={f(\x,\y)=0.6*sin(deg(\x))*cos(deg(\y);}]
        \begin{axis}[scale=2,axis equal,view={225}{15},axis lines=center,axis on top,zmax=2.5,zmin=-1,samples=30,xlabel={$X$},ylabel={$Y$},zlabel={$F(x,y)$}]%ticks=none,
            \addplot3[surf,color=DarkBlue,opacity=0.2,domain=0:1.5*pi, y domain=0:3,faceted color=black] {f(x,y)};
            %%define P
            \addplot3+ [mark=ball,mark size=2pt,scatter src=rand,ball color=yellow!80!black!60]
            coordinates {({0.7},{1.25},{f(x,y)})};
            %space vector
            \draw[->,green,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})\pgfextra{\computegraf{f}{0.7}{1.25}} --\pgfmathresulta;
            %y vector
            \draw[->,yellow,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})\pgfextra{\computegrag{f}{0.7}{1.25}} --\pgfmathresulta;
            %x vector
            \draw[->,red,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})\pgfextra{\computegrah{f}{0.7}{1.25}} --\pgfmathresulta;
            %z vector
            \draw[->,blue,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})\pgfextra{\computegral{f}{0.7}{1.25}} --\pgfmathresulta;
%%%add lines
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.7},{2.3},{f(0.7,1.25)})--({0.7},{2.3},{1.4+f(0.7,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.3},{1.25},{f(0.7,1.25)})--({0.3},{1.25},{1.4+f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.3},{2.3},{f(0.7,1.25)})--({0.3},{2.3},{1.4+f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.7},{2.3},{1.4+f(0.7,1.25)})--({0.7},{1.25},{1.4+f(0.7,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.3},{2.3},{1.4+f(0.3,1.25)})--({0.3},{1.25},{1.4+f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.3},{2.3},{f(0.3,1.25)})--({0.3},{1.25},{f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.7},{1.25},{1.4+f(0.3,1.25)})--({0.3},{1.25},{1.4+f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.7},{2.3},{1.4+f(0.3,1.25)})--({0.3},{2.3},{1.4+f(0.3,1.25)});
\draw[-,dash pattern=on 4pt off 4pt,violet,line width=1pt]({0.7},{2.3},{f(0.3,1.25)})--({0.3},{2.3},{f(0.3,1.25)});
\end{axis}
\end{tikzpicture}
\end{document}

@U7117 在 \computegrad/h/.. 计算完成之后，把结果保存下来，然后用向量计算公式计算端点的坐标。\fpeval 可以直接计算向量的和。

@u10307 老师，我还是没掌握这种方法，不知代码如何写。要不您有时间再指点一下我。

@U7117

\documentclass[svgnames]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \makeatletter
    \newcommand\computegrad[4][0.00025]{% [delta], function, x, y
        \begingroup\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
        \def\@tempdelta{#1}%
        \pgfmathparse{#2(#3,#4)}\let\@tempz\pgfmathresult
        \pgfmathparse{#2(#3+\@tempdelta,#4)}\let\@temppxa\pgfmathresult
        \pgfmathparse{#2(#3-\@tempdelta,#4)}\let\@temppxb\pgfmathresult
        \edef\@temppx{\fpeval{(\@temppxa-\@temppxb)/(1*\@tempdelta)}}%
        \pgfmathparse{#2(#3,#4+\@tempdelta)}\let\@temppya\pgfmathresult
        \pgfmathparse{#2(#3,#4-\@tempdelta)}\let\@temppyb\pgfmathresult
        \edef\@temppy{\fpeval{(\@temppya-\@temppyb)/(1*\@tempdelta)}}%
        \edef\@tempu{\fpeval{(\@temppx,\@temppy,-1)/sqrt(\@temppx^2+\@temppy^2+1)}}%
        \edef\?{\endgroup\def\noexpand\pgfmathresult{\@tempu}%
            \def\noexpand\pgfmathresulta{\fpeval{1*(#3,#4,\@tempz)-\@tempu}}}%
        \?}
    \def\gradcomponent#1#2#3{%
        \edef\pgfmathresult{\fpeval{#2+\expanded{\noexpand\gradcomponent@#1}{#3}}}}
    \def\gradcomponent@(#1,#2,#3)#4{\ifcase                                                
        \if#4x 1 \else\if#4y 2 \else\if#4z 3 \else 4 \fi\fi\fi
        \or (0,#2,#3)\or (#1,0,#3)\or (#1,#2,0)\else(#1,#2,#3)\fi}
    \def\sumaxis#1#2#3{\fpeval{% 把#1的#3轴替换为#2的#3轴
      \expanded{\noexpand\gradcomponent@#1{#3}}+#2-\expanded{\noexpand\gradcomponent@#2{#3}}}}
    %%%%%%%%%%%%%%%%%green
    \newcommand\computegraf[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}}
    %%%%%%%%%%%%%%%%%yellow
    \newcommand\computegrag[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{y}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%red
    \newcommand\computegrah[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{x}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%blue
    \newcommand\computegral[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
        \let\pgfmathresultb\pgfmathresult
        \gradcomponent\pgfmathresult\pgfmathresulta{z}%
        \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}
    %%%%%%%%%%%%%%%%%blue1
\newcommand\computegrat[4][0.00025]{\computegrad[#1]{#2}{#3}{#4}%
    \let\pgfmathresultb\pgfmathresult
    \gradcomponent\pgfmathresult\pgfmathresulta{z}%
    \let\pgfmathresulta\pgfmathresult \let\pgfmathresult\pgfmathresultb}    
    \makeatother    
    \begin{tikzpicture}
        [declare function={f(\x,\y)=0.6*sin(deg(\x))*cos(deg(\y);}]
        \begin{axis}[scale=2,axis equal,view={225}{15},axis lines=center,axis on top,zmax=2.5,zmin=-1,samples=30,xlabel={$X$},ylabel={$Y$},zlabel={$F(x,y)$}]%ticks=none,
            \addplot3[surf,color=DarkBlue,opacity=0.2,domain=0:1.5*pi, y domain=0:3,faceted color=black] {f(x,y)};
            %%define P
            \addplot3+ [mark=ball,mark size=2pt,scatter src=rand,ball color=yellow!80!black!60]
            coordinates {({0.7},{1.25},{f(x,y)})};
            %space vector
            \computegraf{f}{0.7}{1.25}\let\Vu\pgfmathresulta
            \draw[->,green,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})--\Vu;
            %y vector
            \computegrag{f}{0.7}{1.25}\let\Vy\pgfmathresulta
            \draw[->,yellow,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})--\Vy;
            %x vector
            \computegrah{f}{0.7}{1.25}\let\Vx\pgfmathresulta
            \draw[->,red,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})--\Vx;
            %z vector
            \computegral{f}{0.7}{1.25}\let\Vz\pgfmathresulta
            \draw[->,blue,shorten <=0cm]({0.7},{1.25},{f(0.7,1.25)})--\Vz;
\begin{scope}[dash pattern=on 4pt off 4pt,line width=1pt,color=violet]
\xdef\Vxy{\sumaxis\Vx\Vy y} \xdef\Vxz{\sumaxis\Vx\Vz z} \xdef\Vyz{\sumaxis\Vy\Vz z}
\draw\Vx--\Vxy; \draw\Vx--\Vxz; \draw\Vy--\Vyz; 
\draw\Vy--\Vxy; \draw\Vz--\Vxz; \draw\Vz--\Vyz;
\draw\Vu--\sumaxis\Vu\Vxy z; \draw\Vu--\sumaxis\Vu\Vxz y; \draw\Vu--\sumaxis\Vu\Vyz x;
\end{scope}
\end{axis}
\end{tikzpicture}
\end{document}

@u10307 完美解决，很感激老师。