☰

Eixo 3D e poliedro com segmento de linha em direção à origem

tag-icon • tag-icon tikz-pgf 3d
Eixo 3D e poliedro com segmento de linha em direção à origem

Eu gostaria de fazer um gráfico 3D contendo o eixo 3D e um poliedro convexo ou melhor, um Dodecaedro

captura de tela

no lado positivo para o qual um de seus pontos de borda em seu topo é anotado, por exemplo, \hat{x}"escolhido" ou "selecionado" e desenhe um segmento de linha desse ponto rotulado em direção à origem. Em seguida, destaque e anote também onde esse segmento de linha cruza com a casca convexa do Dodecaedro. Seria um pesadelo fazer isso usando o TikZ?

aliás, qual a melhor maneira de aprender TikZ de uma vez por todas? existem bons livros? Sempre acabo consumindo TikZ de uma forma ou de outra :(

ATUALIZAÇÃO: na verdade preciso de vários gráficos em torno dessa mesma ideia para ilustrar em detalhes o comportamento de um algoritmo que preciso documentar. Espero que vendo como é feito neste caso eu consiga generalizar e fazer os outros sozinho... embora sabendo o quão difícil é TikZ... :(

Responder1

\documentclass{article}
\usepackage[dvipsnames]{pstricks}
\usepackage{pst-solides3d}
\begin{document}

\begin{pspicture}[solidmemory,fontsize=20](-4,-4)(4,4)
\psset{Decran=30,viewpoint=20 40 30 rtp2xyz, lightsrc=viewpoint}
\psSolid[object=dodecahedron,a=2.5,action=draw*,name=my_dodecahedron,
         fillcolor=green!50!white]
\psSolid[object=point,definition=solidgetsommet,
  args=my_dodecahedron 0,linecolor=blue,text=A,pos=uc,name=A]
\psSolid[object=point,definition=solidgetsommet,
  args=my_dodecahedron 4,linecolor=blue,text=B,pos=uc,name=B]
\psSolid[object=line,args=A B,linecolor=blue]
\psSolid[object=vecteur,args=A,linecolor=blue]
\psSolid[object=vecteur,args=B,linecolor=blue]
\axesIIID(2.5,2.5,2.5)(3.5,3,3)
\end{pspicture}
%
\begin{pspicture}[solidmemory,fontsize=20](-4,-4)(4,4)
\psset{Decran=30,viewpoint=20 40 35 rtp2xyz, lightsrc=viewpoint}
\psSolid[object=dodecahedron,a=2.5,action=draw*,RotX=22.5,RotY=22.5,
  fillcolor=red!50!white,name=my_dodecahedron,action=draw**,
%  numfaces=all,num=all,
]
\psSolid[object=point,definition=solidcentreface,
   args=my_dodecahedron 2,linecolor=white,text=Centre face 2,pos=uc]
\psSolid[object=point,definition=solidgetsommet,
  args=my_dodecahedron 0,linecolor=white,text=A,pos=cl,name=A]
\psSolid[object=point,definition=solidgetsommet,
  args=my_dodecahedron 4,linecolor=white,text=B,pos=cl,name=B]
\psSolid[object=line,args=A B,linecolor=white]
\end{pspicture}

\end{document}

insira a descrição da imagem aqui

Responder2

Aqui está umTikZponto de partida, todos os vértices, arestas e faces são definidos de forma independente, para que você possa usá-los para mais magia:

Código

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}

\begin{document}

% golden ratio and inverse golden ratio
\pgfmathsetmacro{\gr}{(1+sqrt(5))/2}
\pgfmathsetmacro{\igr}{2/(1+sqrt(5))}

%choose axis angles
\newcommand{\xangle}{0}
\newcommand{\yangle}{90}
\newcommand{\zangle}{225}

%choose axis lengths
\newcommand{\xlength}{1}
\newcommand{\ylength}{1}
\newcommand{\zlength}{0.5}

\pgfmathsetmacro{\xx}{\xlength*cos(\xangle)}
\pgfmathsetmacro{\xy}{\xlength*sin(\xangle)}
\pgfmathsetmacro{\yx}{\ylength*cos(\yangle)}
\pgfmathsetmacro{\yy}{\ylength*sin(\yangle)}
\pgfmathsetmacro{\zx}{\zlength*cos(\zangle)}
\pgfmathsetmacro{\zy}{\zlength*sin(\zangle)}

\begin{tikzpicture}
[   x={(\xx cm,\xy cm)},
    y={(\yx cm,\yy cm)},
    z={(\zx cm,\zy cm)},
    scale=2,
    every path/.style={thick}
]

% coordinates of the vertices (see wikipedia page)
    % vertices of inscribed cube
    \coordinate (pd1) at (-1,-1,-1);
    \coordinate (pd2) at (-1,-1,1);
    \coordinate (pd3) at (-1,1,-1);
    \coordinate (pd4) at (-1,1,1);
    \coordinate (pd5) at (1,-1,-1);
    \coordinate (pd6) at (1,-1,1);
    \coordinate (pd7) at (1,1,-1);
    \coordinate (pd8) at (1,1,1);
    % "front/back" "outside of cube" points
    \coordinate (pd9) at (0,-\igr,-\gr);
    \coordinate (pd10) at (0,-\igr,\gr);
    \coordinate (pd11) at (0,\igr,-\gr);
    \coordinate (pd12) at (0,\igr,\gr);
    % "top/bottom" "outside of cube" points
    \coordinate (pd13) at (-\igr,-\gr,0);
    \coordinate (pd14) at (-\igr,\gr,0);
    \coordinate (pd15) at (\igr,-\gr,0);
    \coordinate (pd16) at (\igr,\gr,0);
    % "left/right" "outside of cube" points
    \coordinate (pd17) at (-\gr,0,-\igr);
    \coordinate (pd18) at (-\gr,0,\igr);
    \coordinate (pd19) at (\gr,0,-\igr);
    \coordinate (pd20) at (\gr,0,\igr);

% black background rectangle for contrast (better option: backgrounds library)  
    \fill (-2.2,-2) rectangle (2.2,2);

% mark vertices 
    \foreach \x in {1,...,20}
    {   \fill[white] (pd\x) circle (0.03) node[above right] {\tiny\x};
    }

% draw inscribed cube
\draw[gray, densely dotted] (pd8) -- (pd7) -- (pd3) -- (pd4) -- cycle;
\draw[gray, densely dotted] (pd8) -- (pd6) -- (pd5) -- (pd7) -- cycle;
\draw[gray, densely dotted] (pd5) -- (pd6) -- (pd2) -- (pd1) -- cycle;
\draw[gray, densely dotted] (pd1) -- (pd2) -- (pd4) -- (pd3) -- cycle;

% faces; "back" ones gray, "front" ones red
    \fill[gray,fill opacity=0.2] (pd11) -- (pd9) -- (pd5) -- (pd19) -- (pd7) -- cycle;
    \fill[gray,fill opacity=0.2] (pd11) -- (pd9) -- (pd1) -- (pd17) -- (pd3) -- cycle;
    \fill[gray,fill opacity=0.2] (pd11) -- (pd7) -- (pd16) -- (pd14) -- (pd3) -- cycle;
    \fill[gray,fill opacity=0.2] (pd3) -- (pd14) -- (pd4) -- (pd18) -- (pd17) -- cycle;
    \fill[gray,fill opacity=0.2] (pd1) -- (pd9) -- (pd5) -- (pd15) -- (pd13) -- cycle;
    \fill[gray,fill opacity=0.2] (pd1) -- (pd13) -- (pd2) -- (pd18) -- (pd17) -- cycle;
    \fill[red,fill opacity=0.2] (pd14) -- (pd16) -- (pd8) -- (pd12) -- (pd4) -- cycle;
    \fill[red,fill opacity=0.2] (pd8) -- (pd16) -- (pd7) -- (pd19) -- (pd20) -- cycle;
    \fill[red,fill opacity=0.2] (pd20) -- (pd19) -- (pd5) -- (pd15) -- (pd6) -- cycle;
    \fill[red,fill opacity=0.2] (pd12) -- (pd8) -- (pd20) -- (pd6) -- (pd10) -- cycle;
    \fill[red,fill opacity=0.2] (pd10) -- (pd6) -- (pd15) -- (pd13) -- (pd2) -- cycle;
    \fill[red,fill opacity=0.2] (pd12) -- (pd10) -- (pd2) -- (pd18) -- (pd4) -- cycle;

% edges on "back"    face of inscribes cube
    \draw[red] (pd9) -- (pd11);
    \draw[red] (pd11) -- (pd3);
    \draw[red] (pd11) -- (pd7);
    \draw[red] (pd9) -- (pd1);
    \draw[red] (pd9) -- (pd5);
% edges on "top"     face of inscribes cube
    \draw[blue] (pd14) -- (pd16);
    \draw[blue] (pd16) -- (pd8);
    \draw[blue] (pd16) -- (pd7);
    \draw[blue] (pd14) -- (pd3);
    \draw[blue] (pd14) -- (pd4);
% edges on "left"    face of inscribes cube
    \draw[green] (pd17) -- (pd18);
    \draw[green] (pd17) -- (pd3);
    \draw[green] (pd17) -- (pd1);
    \draw[green] (pd18) -- (pd2);
    \draw[green] (pd18) -- (pd4);
% edges on "bottom"  face of inscribes cube
    \draw[yellow] (pd13) -- (pd15);
    \draw[yellow] (pd13) -- (pd1);
    \draw[yellow] (pd13) -- (pd2);
    \draw[yellow] (pd15) -- (pd5);
    \draw[yellow] (pd15) -- (pd6);
% edges on "front"   face of inscribes cube
    \draw[violet] (pd10) -- (pd12);
    \draw[violet] (pd12) -- (pd4);
    \draw[violet] (pd12) -- (pd8);
    \draw[violet] (pd10) -- (pd2);
    \draw[violet] (pd10) -- (pd6);
% edges on "right"   face of inscribes cube 
    \draw[orange] (pd20) -- (pd19); 
    \draw[orange] (pd19) -- (pd7);
    \draw[orange] (pd19) -- (pd5);
    \draw[orange] (pd20) -- (pd8);
    \draw[orange] (pd20) -- (pd6);
\end{tikzpicture}

\end{document}

Resultado

insira a descrição da imagem aqui

Editar 1:Existem vários problemas ao fazer isso emTikZ, pois também os pontos 3D são armazenados internamente em pontos 2D. Além disso, você não consegue encontrar linhas ocultas automaticamente, então você mesmo terá que fazer isso. Com o problema que você descreveu, haveria o problema de saber por qual das 12 superfícies passa a linha de conexão, então escolhi uma onde seja fácil de ver. A macro que escrevi para determinar a interseção só funciona se sua linha estiver passando pela origem.

Código

\documentclass[tikz]{standalone}
\usepackage{xifthen}

\begin{document}

%command to find intersection of plane through abc and line p (through origin)
\newcommand{\planelineinter}[5]% a, b, c, p as {a_x,a_y,a_z}, coordinate name
{   \foreach \a [count=\k] in {#1}
    { \ifthenelse{\k=1}{\xdef\tempxa{\a}}
        \ifthenelse{\k=2}{\xdef\tempya{\a}}
        \ifthenelse{\k=3}{\xdef\tempza{\a}}
    }
    \foreach \b [count=\k] in {#2}
    { \ifthenelse{\k=1}{\xdef\tempxb{\b}}
        \ifthenelse{\k=2}{\xdef\tempyb{\b}}
        \ifthenelse{\k=3}{\xdef\tempzb{\b}}
    }
    \foreach \c [count=\k] in {#3}
    { \ifthenelse{\k=1}{\xdef\tempxc{\c}}
        \ifthenelse{\k=2}{\xdef\tempyc{\c}}
        \ifthenelse{\k=3}{\xdef\tempzc{\c}}
    }
    \foreach \p [count=\k] in {#4}
    { \ifthenelse{\k=1}{\xdef\tempxp{\p}}
        \ifthenelse{\k=2}{\xdef\tempyp{\p}}
        \ifthenelse{\k=3}{\xdef\tempzp{\p}}
    }
    \pgfmathsetmacro{\abx}{\tempxb-\tempxa}
    \pgfmathsetmacro{\aby}{\tempyb-\tempya}
    \pgfmathsetmacro{\abz}{\tempzb-\tempza}
    \pgfmathsetmacro{\acx}{\tempxc-\tempxa}
    \pgfmathsetmacro{\acy}{\tempyc-\tempya}
    \pgfmathsetmacro{\acz}{\tempzc-\tempza}
    \pgfmathsetmacro{\nx}{\aby*\acz-\abz*\acy}
    \pgfmathsetmacro{\ny}{\abz*\acx-\abx*\acz}
    \pgfmathsetmacro{\nz}{\abx*\acy-\aby*\acx}
    \pgfmathsetmacro{\d}{(\nx+\ny+\nz)/(\nx*\tempxp+\ny*\tempyp+\nz*\tempzp)}
    \path (0,0,0) -- (#4) coordinate[pos=\d] (#5);
}

% golden ratio and inverse golden ratio
\pgfmathsetmacro{\gr}{(1+sqrt(5))/2}
\pgfmathsetmacro{\igr}{2/(1+sqrt(5))}

%choose axis angles
\newcommand{\xangle}{0}
\newcommand{\yangle}{90}
\newcommand{\zangle}{225}

%choose axis lengths
\newcommand{\xlength}{1}
\newcommand{\ylength}{1}
\newcommand{\zlength}{0.5}

\pgfmathsetmacro{\xx}{\xlength*cos(\xangle)}
\pgfmathsetmacro{\xy}{\xlength*sin(\xangle)}
\pgfmathsetmacro{\yx}{\ylength*cos(\yangle)}
\pgfmathsetmacro{\yy}{\ylength*sin(\yangle)}
\pgfmathsetmacro{\zx}{\zlength*cos(\zangle)}
\pgfmathsetmacro{\zy}{\zlength*sin(\zangle)}

\begin{tikzpicture}
[   x={(\xx cm,\xy cm)},
  y={(\yx cm,\yy cm)},
  z={(\zx cm,\zy cm)},
  scale=2,
  every path/.style={thick}
]

% coordinates of the vertices (see wikipedia page)
    \node[below left] at (0,0,0) {$\vec{0}$};
    \fill (0,0,0) circle (0.03);
    % vertices of inscribed cube
    \coordinate (pd1) at (-1,-1,-1);
    \coordinate (pd2) at (-1,-1,1);
    \coordinate (pd3) at (-1,1,-1);
    \coordinate (pd4) at (-1,1,1);
    \coordinate (pd5) at (1,-1,-1);
    \coordinate (pd6) at (1,-1,1);
    \coordinate (pd7) at (1,1,-1);
    \coordinate (pd8) at (1,1,1);
    % "front/back" "outside of cube" points
    \coordinate (pd9) at (0,-\igr,-\gr);
    \coordinate (pd10) at (0,-\igr,\gr);
    \coordinate (pd11) at (0,\igr,-\gr);
    \coordinate (pd12) at (0,\igr,\gr);
    % "top/bottom" "outside of cube" points
    \coordinate (pd13) at (-\igr,-\gr,0);
    \coordinate (pd14) at (-\igr,\gr,0);
    \coordinate (pd15) at (\igr,-\gr,0);
    \coordinate (pd16) at (\igr,\gr,0);
    % "left/right" "outside of cube" points
    \coordinate (pd17) at (-\gr,0,-\igr);
    \coordinate (pd18) at (-\gr,0,\igr);
    \coordinate (pd19) at (\gr,0,-\igr);
    \coordinate (pd20) at (\gr,0,\igr);

% ========== the point of interest, part 1
    \coordinate (x) at (4,3,0);
    \planelineinter{1,1,-1}{1,1,1}{\igr,\gr,0}{4,3,0}{interpoint}
    \draw[very thick,red,densely dashed] (0,0) -- (interpoint);

% faces; "back" ones gray, "front" ones red
    \fill[gray,fill opacity=0.4] (pd11) -- (pd9) -- (pd5) -- (pd19) -- (pd7) -- cycle;
    \fill[gray,fill opacity=0.4] (pd11) -- (pd9) -- (pd1) -- (pd17) -- (pd3) -- cycle;
    \fill[gray,fill opacity=0.4] (pd11) -- (pd7) -- (pd16) -- (pd14) -- (pd3) -- cycle;
    \fill[gray,fill opacity=0.4] (pd3) -- (pd14) -- (pd4) -- (pd18) -- (pd17) -- cycle;
    \fill[gray,fill opacity=0.4] (pd1) -- (pd9) -- (pd5) -- (pd15) -- (pd13) -- cycle;
    \fill[gray,fill opacity=0.4] (pd1) -- (pd13) -- (pd2) -- (pd18) -- (pd17) -- cycle;

    \fill[gray,fill opacity=0.4] (pd14) -- (pd16) -- (pd8) -- (pd12) -- (pd4) -- cycle;
    \fill[lime,fill opacity=0.4] (pd8) -- (pd16) -- (pd7) -- (pd19) -- (pd20) -- cycle;
    \fill[gray,fill opacity=0.4] (pd20) -- (pd19) -- (pd5) -- (pd15) -- (pd6) -- cycle;
    \fill[gray,fill opacity=0.4] (pd12) -- (pd8) -- (pd20) -- (pd6) -- (pd10) -- cycle;
    \fill[gray,fill opacity=0.4] (pd10) -- (pd6) -- (pd15) -- (pd13) -- (pd2) -- cycle;
    \fill[gray,fill opacity=0.4] (pd12) -- (pd10) -- (pd2) -- (pd18) -- (pd4) -- cycle;

% edges on "back"    face of inscribes cube; red
    \draw[dashed] (pd9) -- (pd11);
    \draw[dashed] (pd11) -- (pd3);
    \draw[dashed] (pd11) -- (pd7);
    \draw[dashed] (pd9) -- (pd1);
    \draw[dashed] (pd9) -- (pd5);
% edges on "top"     face of inscribes cube
    \draw[] (pd14) -- (pd16);
    \draw[] (pd16) -- (pd8);
    \draw[] (pd16) -- (pd7);
    \draw[dashed] (pd14) -- (pd3);
    \draw[] (pd14) -- (pd4);
% edges on "left"    face of inscribes cube
    \draw[dashed] (pd17) -- (pd18);
    \draw[dashed] (pd17) -- (pd3);
    \draw[dashed] (pd17) -- (pd1);
    \draw[] (pd18) -- (pd2);
    \draw[] (pd18) -- (pd4);
% edges on "bottom"  face of inscribes cube
    \draw[] (pd13) -- (pd15);
    \draw[dashed] (pd13) -- (pd1);
    \draw[] (pd13) -- (pd2);
    \draw[] (pd15) -- (pd5);
    \draw[] (pd15) -- (pd6);
% edges on "front"   face of inscribes cube
    \draw[] (pd10) -- (pd12);
    \draw[] (pd12) -- (pd4);
    \draw[] (pd12) -- (pd8);
    \draw[] (pd10) -- (pd2);
    \draw[] (pd10) -- (pd6);
% edges on "right"   face of inscribes cube 
    \draw[] (pd20) -- (pd19);   
    \draw[] (pd19) -- (pd7);
    \draw[] (pd19) -- (pd5);
    \draw[] (pd20) -- (pd8);
    \draw[] (pd20) -- (pd6);    

% ========== the point of interest, part 2
    \draw[very thick,red] (interpoint) -- (x);
    \fill[blue] (x) circle (0.03) node[above] {$\mathbf{\hat{x}}$};
    \fill[blue] (interpoint) circle (0.03) node[above,fill,white,rounded corners=1mm,fill opacity=0.5,text opacity=1,text=black,above left=1mm] {intersection point};
\end{tikzpicture}

\end{document}

Saída

insira a descrição da imagem aqui

informação relacionada