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3Dドローンに角度を追加する

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3Dドローンに角度を追加する

図のように、オイラー角と XYZ 角度をコードに追加するにはどうすればよいですか?ここに画像の説明を入力してください

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{3d, arrows.meta, calc, quotes}
\begin{document}
\begin{tikzpicture}[
  x = ( 20:-2cm), y = (-30: 2cm), z = ( 80: 1.6cm),
  > = Triangle]
\draw coordinate (O) (0,0,-.05) ellipse [radius=.2];

\foreach \xy/\yx in {x/y, y/x} {
  \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
            -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
  \foreach \lmr in {-.05, 0, .05}
    \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
  \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                (xyz cs: \xy =  1) ellipse[];
}
\draw[fill=white] (O) ellipse [radius=.2];

\foreach \xyz/\Label/\Ang in {% \Ang needs to start with a ,
  x/left/ {, longitude = 70},
  y/right/{, longitude =-20},
  z/above/{, longitude = 70, latitude=70}%
} {
  \draw[->, densely dashed, red] (0, 0, 0) -- ++(xyz cs: \xyz= 1.5)
    coordinate (ax-\xyz) node[\Label] {$\xyz$};
  \draw[blue, ->]
     (0, 0, 0) -- (xyz spherical cs: radius = 1.5,/tikz/style/.expanded=\Ang)
      coordinate (ax-\xyz') node[\Label] {$\xyz$};
}

\draw[canvas is xy plane at z=0]
  (0, 1) arc[start angle=90, delta angle=20, radius=1]
  node[midway, below right] {$\varphi$};
\end{tikzpicture}
\end{document}

答え1

パッケージの使用をお勧めしますtikz-3dplot。パッケージ内には、オイラー角を使用した座標変換がすでに定義されています。同様の質問と役立つ回答が見つかります。ここ基本的に、2 つの角度 (例 ) で方向を設定できるメイン座標系を定義できます\tdplotsetmaincoords{50}{115}。次に、この座標系をコマンドで回転できます\tdplotsetrotatedcoords{<angle1>}{<angle2>}{<angle3>}。パッケージ内では、このコマンドは、シーケンス zyz のオイラー角を使用して座標系の回転を実行するように実装されています。

別のシーケンスを取得するには、すでに説明したように変換コマンドを再定義します。これそしてこの答えに前述の質問この方法では、最終的にすべての可能なシーケンスを追加し、それらを切り替えることができます。以下のコードでは、zyxシーケンス（Tait-BryanまたはCardan角度と呼ばれます（オイラー角に関するWikipedia)) は、与えられた図に最も近いだけでなく、Tait-Byran 角や、最も一般的な古典的なオイラー角としての zxz シーケンスでも最も一般的なため、この方法を採用しました。

その後、メイン座標系または回転座標系のいずれかで描画できます。この方法では、ドローンなどの 3D 描画や、角度を示す円弧を簡単に追加できます。tikz-3dplotパッケージには\tdplotdrawarc[coordinates_frame,->,colo]{origin}{radius}{start angle}{end angle}{node info}{label}コマンドが含まれていますが、既に使用したオプションを使用して、回転座標系の平面に円弧を手動で描画する方が直感的であることがわかりましたcanvas is xy plane at z=0。

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, quotes}
%\usetikzlibrary{3d, arrows.meta, calc, quotes}

\usepackage{tikz-3dplot}

% Define new arc command
% Syntax: [draw options] (center) (initial angle:final angle:radius)
% https://tex.stackexchange.com/a/66220
\def\centerarc[#1](#2)(#3:#4:#5){%
    \draw [#1] ($(#2)+({#5*cos(#3)}, {#5*sin(#3)})$) arc (#3:#4:#5)%
}
 
% Redefine the rotation sequence for the tikz3d-plot to Euler-Angles:
% z-x-z with alpha-beta-gamma (psi-theta-phi)
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdseteulerzxz}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sacbsg}{\sacb}{\singamma}
        \tdplotmult{\sacbcg}{\sacb}{\cosgamma}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\sbsg}{\sinbeta}{\singamma}
        \tdplotmult{\sbcg}{\sinbeta}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\cacbsg}{\cacb}{\singamma}
        \tdplotmult{\cacbcg}{\cacb}{\cosgamma}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacg - \sacbsg}
        \pgfmathsetmacro{\rabeul}{-\casg - \sacbcg}
        \pgfmathsetmacro{\raceul}{+\sasb}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacg + \cacbsg}
        \pgfmathsetmacro{\rbbeul}{-\sasg + \cacbcg}
        \pgfmathsetmacro{\rbceul}{-\casb}
        %
        \pgfmathsetmacro{\rcaeul}{+\sbsg}
        \pgfmathsetmacro{\rcbeul}{+\sbcg}
        \pgfmathsetmacro{\rcceul}{+\cosbeta}
        %
    }
}
 
% Redefine the rotation sequence for the tikz3d-plot to Cardan-Angles:
% z-y-x with alpha-beta-gamma
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdsetcardanzyx}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sasbsg}{\sasb}{\singamma}
        \tdplotmult{\sasbcg}{\sasb}{\cosgamma}
        %
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\casbsg}{\casb}{\singamma}
        \tdplotmult{\casbcg}{\casb}{\cosgamma}
        %
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        \tdplotmult{\cbsg}{\cosbeta}{\singamma}
        \tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacb}
        \pgfmathsetmacro{\rabeul}{+\casbsg - \sacg}
        \pgfmathsetmacro{\raceul}{+\sasg+\casbcg}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacb}
        \pgfmathsetmacro{\rbbeul}{+\cacg + \sasbsg}
        \pgfmathsetmacro{\rbceul}{\sasbcg-\casg}
        %
        \pgfmathsetmacro{\rcaeul}{-\sinbeta}
        \pgfmathsetmacro{\rcbeul}{+\cbsg}
        \pgfmathsetmacro{\rcceul}{+\cbcg}
        %
    }
}
 
% Plot display orientation
\tdplotsetmaincoords{50}{115}

\begin{document}


% z-y-x Tait-Bryan angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdsetcardanzyx
    
    \pgfmathsetmacro{\angpsi}{-20}  % or alpha
    \pgfmathsetmacro{\angtheta}{-10}    % or beta
    \pgfmathsetmacro{\angphi}{-5}   % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new y-axis
        \draw[canvas is xz plane at y=0,->] (1.5,0) arc (0:-\angtheta:1.5);
        \node at ({1.7*cos(\angtheta/2)},0,{-1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is xz plane at y=0,->] (0,1.5) arc (90:90-\angtheta:1.5);
        \node at ({1.7*sin(\angtheta/2)},0,{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {z/above}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at (0,{1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)}){$\varphi$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at (0,{-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)}){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}

% z-x-z Euler angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdseteulerzxz
    
    \pgfmathsetmacro{\angpsi}{-10}  % or alpha
    \pgfmathsetmacro{\angtheta}{-20}    % or beta
    \pgfmathsetmacro{\angphi}{-10}  % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angtheta:1.5);
        \node at (0,{1.7*cos(\angtheta/2)},{1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angtheta:1.5);
        \node at (0,{-1.7*sin(\angtheta/2)},{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {y/right}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new z-axis
        \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at ({1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)},0){$\varphi$};
        \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at ({-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)},0){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}


\end{document}

結果：

テイト・ブライアン・ジックス テイト-ブライアン_zyx オイラーzxz オイラー

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