# What is the advantage of a transformation matrix in perspective projection?

I was messing around with perspective projection. And it got me wondering whether using a perspective transformation matrix is the most efficient way. Because I also came up with another method which seems to need about the same amount of calculations:

Given:

• The position of the "camera" $\vec{O}$.
• The normalized vector $\vec{n}$ in the direction in which the camera is looking (normal of the plane on which the points will be projected).
• Vectors $\vec{X}$ and $\vec{Y}$ which determine the orientation of the projection. Both should lie in the projection plane and together with $\vec{n}$ should span an orthonormal basis in the $\mathbb{R}^3$.
• The shortest distance from the camera to projection plane $h$
• The point $\vec{P}$ which needs to be projected.

$\vec{\Delta}=\vec{P}-\vec{O}$ with $\vec{\Delta}$ the relative coordinates of $\vec{P}$ with respect to $\vec{O}$.

$\vec{\alpha}=h\left(\frac{\vec{\Delta}}{\vec{\Delta}\cdot\vec{n}}-\vec{n}\right)$

$x=\vec{\alpha}\cdot\vec{X}$

$y=\vec{\alpha}\cdot\vec{Y}$

Or do methods, which use transformation matrices, use eigen-value/vectors to reduce the amount of calculations?