3D Modeling - Using Homogeneous Coordinates

Using Homogeneous Coordinates

In projective geometry, often used in computer graphics, points are represented using homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py, pz, 1) would need to be multiplied with this projective transformation matrix:

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}.

As shown below, the multiplication will give the expected result:

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}.

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:

S_v = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{s} \end{bmatrix}.

For each vector p = (px, py, pz, 1) we would have

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}

which would be homogenized to

\begin{bmatrix} sp_x \\ sp_y \\ sp_z \\ 1 \end{bmatrix}.

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