Parallel Projection

Resources:

• page=5
• page=6

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Parallel projection, where the distance of the Center of Projection from the plane of projection is infinite (meaning that the projector lines are parallell).

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Examples

Orthographic Projection

By using one of the main planes as the plane of projection (in this case, the $xy$ plane), we can project a point $\mathbf{P}=[x,y,z{]}^T$ onto $[x^{\prime },y^{\prime },z^{\prime }{]}^T=[x,y,0{]}^T$ by using the following matrix

$${\mathbf{P}}_{\text{ORTHO}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

Such that ${\mathbf{P}}^{\prime }={\mathbf{P}}_{\text{ORTHO}}\cdot \mathbf{P}$

Oblique Projection

Let the direction of projection be given by $\overrightarrow{DOP}=[DO{P}_x,DO{P}_y,DO{P}_z{]}^T$, and we want to project this onto a given plane (in this case we use the $xz$-plane). Then the projection matrix is given by:

$${\mathbf{P}}_{\text{OBLIQUE}}(\overrightarrow{DOP})=\left[\begin{array}{cccc}1& 0& -\frac{DO{P}_x}{DO{P}_z}& 0\\ 0& 1& -\frac{DO{P}_y}{DO{P}_z}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

See page 7 in the book for the proof