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

Famous quotes containing the word mathematics:

    “Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the mathematics of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.”
    —Walter Reisch (1903–1963)

    “It is a monstrous thing to force a child to learn Latin or Greek or mathematics on the ground that they are an indispensable gymnastic for the mental powers. It would be monstrous even if it were true.”
    —George Bernard Shaw (1856–1950)

    “Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don’t happen to have all the data. In mathematics we have all the data ... and yet we don’t understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.”
    —Simone Weil (1909–1943)