Editing Isometric projection (section)

==Mathematics==
There are eight different orientations to obtain an isometric view, depending into which [[octant (solid geometry)|octant]] the viewer looks. The isometric transform from a point ''a''{{sub|''x'',''y'',''z''}} in 3D space to a point ''b''{{sub|''x'',''y''}} in 2D space looking into the first octant can be written mathematically with [[rotation matrix|rotation matrices]] as:
<math display="block">
\begin{bmatrix}
   \mathbf{c}_x \\
   \mathbf{c}_y \\
   \mathbf{c}_z \\
\end{bmatrix}=\begin{bmatrix}
   1 & 0 & 0  \\
   0 & {\cos\alpha} & {\sin\alpha}  \\
   0 & { - \sin\alpha} & {\cos\alpha}  \\
\end{bmatrix}\begin{bmatrix}
   {\cos\beta } & 0 & { - \sin\beta }  \\
   0 & 1 & 0  \\
   {\sin\beta } & 0 & {\cos\beta }  \\
\end{bmatrix}\begin{bmatrix}
   \mathbf{a}_x \\
   \mathbf{a}_y \\
   \mathbf{a}_z \\
\end{bmatrix}=\frac{1}{\sqrt{6}}\begin{bmatrix}
   \sqrt{3} & 0 & -\sqrt{3}  \\
   1 & 2 & 1  \\
   \sqrt{2} & -\sqrt{2} & \sqrt{2}  \\
\end{bmatrix}\begin{bmatrix}
   \mathbf{a}_x \\
   \mathbf{a}_y \\
   \mathbf{a}_z \\
\end{bmatrix}
</math>

where ''α'' = arcsin(tan 30°) ≈ 35.264° and ''β'' = 45°. As explained above, this is a rotation around the vertical (here ''y'') axis by ''β'', followed by a rotation around the horizontal (here ''x'') axis by ''α''. This is then followed by an orthographic projection to the ''xy''-plane:
<math display="block">
\begin{bmatrix}
   \mathbf{b}_x \\
   \mathbf{b}_y \\
   0 \\
\end{bmatrix} =
\begin{bmatrix}
   1 & 0 & 0  \\
   0 & 1 & 0  \\
   0 & 0 & 0  \\
\end{bmatrix}\begin{bmatrix}
   \mathbf{c}_x \\
   \mathbf{c}_y \\
   \mathbf{c}_z \\
\end{bmatrix}
</math>

The other 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.<ref>{{cite journal
|author1=Ingrid Carlbom |author2=Joseph Paciorek |author3=Dan Lim |title=Planar Geometric Projections and Viewing Transformations
|journal=[[ACM Computing Surveys]]
|volume=10|issue=4
|pages=465–502
|date=December 1978
|doi=10.1145/356744.356750
|citeseerx=10.1.1.532.4774 |s2cid=708008 }}</ref>