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PGF/TikZ Manual

TikZ and PGF Manual

Libraries

63 Three Point Perspective Drawing Library

by Max Snippe

  • TikZ Library perspective

    • \usetikzlibrary{perspective} % and plain
    \usetikzlibrary[perspective] % Cont

    This library provides tools for perspective drawing with one, two, or three vanishing points.

63.1 Coordinate Systems

  • Coordinate system three point perspective

    • The three point perspective coordinate system is very similar to the xyz coordinate system, save that it will display the provided coordinates with a perspective projection.

    • /tikz/cs/x=⟨number⟩ (no default, initially 0)

      • The \(x\) component of the coordinate. Should be given without unit.

    • /tikz/cs/y=⟨number⟩ (no default, initially 0)

      • Same as x.

    • /tikz/cs/z=⟨number⟩ (no default, initially 0)

      • Same as x.

  • Coordinate system tpp

    • The tpp coordinate system is an alias for the three point perspective coordinate system.

63.2 Setting the view

  • /tikz/3d view={⟨azimuth⟩}{⟨elevation⟩} (default {-30}{15})

    • With the 3d view option, the projection of the 3D coordinates on the 2D page is defined. It is determined by rotating the coordinate system by \(-\meta {azimuth}\) around the \(z\)-axis, and by ⟨elevation⟩ around the (new) \(x\)-axis, as shown below.

    (-tikz- diagram)

    For example, when both ⟨azimuth⟩ and ⟨elevation⟩ are 0\(^\circ \), \(+z\) will be pointing upward, and \(+x\) will be pointing right. The default is as shown below.

    (-tikz- diagram)

    \usetikzlibrary {perspective}
    \begin{tikzpicture}[3d view]
    \draw[->] (-1,0,0) -- (1,0,0) node[pos=1.1]{x};
    \draw[->] (0,-1,0) -- (0,1,0) node[pos=1.1]{y};
    \draw[->] (0,0,-1) -- (0,0,1) node[pos=1.1]{z};
    \end{tikzpicture}

  • /tikz/isometric view(style, no value)

    • A special kind of 3d view is isometric, which can be set with the isometric view style. It simply sets 3d view={-45}{35.26}. The value for ⟨elevation⟩ is determined with \(\arctan (1/\sqrt {2})\). In isometric projection the angle between any pair of axes is 120\(^\circ \), as shown below.

    (-tikz- diagram)

    \usetikzlibrary {perspective}
    \begin{tikzpicture}[isometric view]
    \draw[->] (-1,0,0) -- (1,0,0) node[pos=1.1]{x};
    \draw[->] (0,-1,0) -- (0,1,0) node[pos=1.1]{y};
    \draw[->] (0,0,-1) -- (0,0,1) node[pos=1.1]{z};
    \end{tikzpicture}
63.3 Defining the perspective

In this section, the following example cuboid will be used with various scaling. As a reference, the axes will be shown too, without perspective projection.

(-tikz- diagram)

\usetikzlibrary {perspective}
\newcommand\simplecuboid[3]{%
\fill[gray!80!white] (tpp cs:x=0,y=0,z=#3)
-- (tpp cs:x=0,y=#2,z=#3)
-- (tpp cs:x=#1,y=#2,z=#3)
-- (tpp cs:x=#1,y=0,z=#3) -- cycle;
\fill[gray] (tpp cs:x=0,y=0,z=0)
-- (tpp cs:x=0,y=0,z=#3)
-- (tpp cs:x=0,y=#2,z=#3)
-- (tpp cs:x=0,y=#2,z=0) -- cycle;
\fill[gray!50!white] (tpp cs:x=0,y=0,z=0)
-- (tpp cs:x=0,y=0,z=#3)
-- (tpp cs:x=#1,y=0,z=#3)
-- (tpp cs:x=#1,y=0,z=0) -- cycle;}
\newcommand{\simpleaxes}[3]{%
\draw[->] (-0.5,0,0) -- (#1,0,0) node[pos=1.1]{x};
\draw[->] (0,-0.5,0) -- (0,#2,0) node[pos=1.1]{y};
\draw[->] (0,0,-0.5) -- (0,0,#3) node[pos=1.1]{z};}

\begin{tikzpicture}[3d view]
\simplecuboid{2}{2}{2}
\simpleaxes{2}{2}{2}
\end{tikzpicture}

  • /tikz/perspective=⟨vanishing points⟩ (default p={(10,0,0)},q={(0,10,0)},r={(0,0,20)})

    • The ‘strength’ of the perspective can be determined by setting the location of the vanishing points. The default values have a stronger perspective towards \(x\) and \(y\) than towards \(z\), as shown below.

    (-tikz- diagram)

    \usetikzlibrary {perspective}
    \begin{tikzpicture}[3d view,perspective]
    \simplecuboid{2}{2}{2}
    \simpleaxes{2}{2}{2}
    \end{tikzpicture}

    From this example it also shows that the maximum dimensions of the cuboid are no longer 2 by 2 by 2. This is inherent to the perspective projection.

    • /tikz/perspective/p={⟨x,y,z⟩} (no default, initially (0,0,0))

      • The location of the vanishing point that determines the ‘strength’ of the perspective in \(x\)-direction can be set with the p key.

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      p = {(5,0,0)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}

      Note also that when only p is provided, the perspective in \(y\) and \(z\) direction is turned off.

      To turn off the perspective in \(x\)-direction, one must set the \(x\) component of p to 0 (e.g. p={(0,a,b)}, where a and b can be any number and will be ignored). Or one can provide q and r and omit p.

      By changing the \(y\) and \(z\) components of p, one can achieve various effects.

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      p = {(5,0,1)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      p = {(5,1,0)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      p = {(5,1,1)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}

    • /tikz/perspective/q={⟨x,y,z⟩} (no default, initially (0,0,0))

      • Similar to p, but can be turned off by setting its \(y\) component to 0.

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      q = {(0,5,0)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}

    • /tikz/perspective/r={⟨x,y,z⟩} (no default, initially (0,0,0))

      • Similar to p, but can be turned off by setting its \(z\) component to 0.

      (-tikz- diagram)

      \usetikzlibrary {perspective}
      \begin{tikzpicture}[
      3d view,
      perspective={
      r = {(0,0,5)}}      ]
      \simplecuboid{2}{2}{2}
      \simpleaxes{2}{2}{2}
      \end{tikzpicture}
63.4 Shortcomings

Currently a number of things are not working, mostly due to the fact that PGF uses a 2D coordinate system underwater, and perspective projection is a non-linear affine transformation which needs to be aware of all three coordinates. These three coordinates are currently lost when processing a 3D coordinate. The issues include, but possibly are not limited to:

  • • Keys like shift, xshift, yshift are not working

  • • Keys like rotate around x, rotate around y, and rotate around z are not working

  • • Units are not working

  • • Most keys from the 3d library are unsupported, e.g. all the canvas is .. plane keys.

63.5 Examples

An r that lies ‘below’ your drawing can mimic a macro effect.

(-tikz- diagram)

\usetikzlibrary {perspective}
\begin{tikzpicture}[
isometric view,
perspective={
p = {(8,0,0)},
q = {(0,8,0)},
r = {(0,0,-8)}}]

\simplecuboid{2}{2}{2}

\end{tikzpicture}

A peculiar phenomenon inherent to perspective drawing, is that however great your coordinate will become in the direction of the vanishing point, it will never reach it.

(-tikz- diagram)

\usetikzlibrary {perspective}
\begin{tikzpicture}[
isometric view,
perspective={
p = {(4,0,0)},
q = {(0,4,0)}}]

\node[fill=red,circle,inner sep=1.5pt,label=above:p] at (4,0,0){};

\foreach \i in {0,...,100}{
\filldraw[fill = gray] (tpp cs:x=\i,y=0,z=0)
-- (tpp cs:x=\i+0.5,y=0,z=0)
-- (tpp cs:x=\i+0.5,y=2,z=0)
-- (tpp cs:x=\i,y=2,z=0)
-- cycle;}
\end{tikzpicture}

Even for simple examples, the added perspective might add another ‘dimension’ to your drawing. In this case, two vanishing points give a more intuitive result then three would.

(-tikz- diagram)

\usetikzlibrary {perspective}
\begin{tikzpicture}[
scale=0.7,
3d view,
perspective={
p = {(20,0,0)},
q = {(0,20,0)}}]

\filldraw[fill=brown] (tpp cs:x=0,y=0,z=0)
-- (tpp cs:x=0,y=4,z=0)
-- (tpp cs:x=0,y=4,z=2)
-- (tpp cs:x=0,y=2,z=4)
-- (tpp cs:x=0,y=0,z=2) -- cycle;
\filldraw[fill=red!70!black] (tpp cs:x=0,y=0,z=2)
-- (tpp cs:x=5,y=0,z=2)
-- (tpp cs:x=5,y=2,z=4)
-- (tpp cs:x=0,y=2,z=4) -- cycle;
\filldraw[fill=brown!80!white] (tpp cs:x=0,y=0,z=0)
-- (tpp cs:x=0,y=0,z=2)
-- (tpp cs:x=5,y=0,z=2)
-- (tpp cs:x=5,y=0,z=0) -- cycle;
\end{tikzpicture}

With the vanishing points nearby, the distortion of parallel lines becomes very strong. This might lead to Dimension too large errors.

(-tikz- diagram)

\usetikzlibrary {perspective}
\begin{tikzpicture}[
3d view,
perspective={
p = {(2,0,0)},
q = {(0,2,0)},
r = {(0,0,2)}},
scale=4,
vanishing point/.style={fill,circle,inner sep=2pt}]

\simplecuboid{3}{1}{2}

\node[vanishing point,label = right:p] (p) at (2,0,0){};
\node[vanishing point,label = left:q] (q) at (0,2,0){};
\node[vanishing point,label = above:r] (r) at (0,0,2){};

\begin{scope}[dotted]
\foreach \y in {0,1}{
\foreach \z in {0,2}{
\draw (tpp cs:x=0,y=\y,z=\z) -- (p.center);}}
\foreach \x in {0,3}{
\foreach \z in {0,2}{
\draw (tpp cs:x=\x,y=0,z=\z) -- (q.center);}}
\foreach \x in {0,3}{
\foreach \y in {0,1}{
\draw (tpp cs:x=\x,y=\y,z=0) -- (r.center);}}
\end{scope}
\end{tikzpicture}