Isometric Projection - Mathematics

Mathematics

There are eight different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point in 3D space to a point in 2D space looking into the first octant can be written mathematically with rotation matrices as:

\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}

where and . As explained above, this is a rotation around the vertical (here y) axis by, followed by a rotation around the horizontal (here x) axis by . This is then followed by an orthographic projection to the x-y plane:

\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}

The other 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.

Read more about this topic:  Isometric Projection

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