Editing 2D computer graphics (section)

===Using homogeneous coordinates===
In [[projective geometry]], often used in [[computer graphics]], points are represented using [[homogeneous coordinates]]. To scale an object by a [[Vector (geometric)|vector]] ''v'' = (''v<sub>x</sub>, v<sub>y</sub>, v<sub>z</sub>''), each homogeneous coordinate vector ''p'' = (''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>'', 1) would need to be multiplied with this [[projective transformation]] matrix:

:<math> S_v = 
\begin{bmatrix}
v_x & 0 & 0 & 0 \\
0 & v_y & 0 & 0 \\
0 & 0 & v_z & 0 \\
0 & 0 & 0 & 1 
\end{bmatrix}.
</math>

As shown below, the multiplication will give the expected result:
:<math>
S_vp =
\begin{bmatrix}
v_x & 0 & 0 & 0 \\
0 & v_y & 0 & 0 \\
0 & 0 & v_z & 0 \\
0 & 0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ 1 
\end{bmatrix}
=
\begin{bmatrix}
v_xp_x \\ v_yp_y \\ v_zp_z \\ 1 
\end{bmatrix}.
</math>

Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor ''s'' (uniform scaling) can be accomplished by using this scaling matrix:
:<math> S_v = 
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & \frac{1}{s} 
\end{bmatrix}.
</math>

For each vector ''p'' = (''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>'', 1) we would have
:<math>
S_vp =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & \frac{1}{s}  
\end{bmatrix}
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ 1 
\end{bmatrix}
=
\begin{bmatrix}
p_x \\ p_y \\ p_z \\ \frac{1}{s} 
\end{bmatrix}
</math>
which would be homogenized to
:<math>
\begin{bmatrix}
sp_x \\ sp_y \\ sp_z \\ 1 
\end{bmatrix}.
</math>