Geometric Interpretation of Homogeneous Coordinates

Transformation matrix 3×3 for two-dimensional homogeneous coordinates can be divided into four parts:

(1.54)

Recall that a, b, c and d are coefficients of the scaling, rotation, reflection, and shear, respectively. Elements m and n define the movement. Let us set that the values ​​of p and q are not equal to 0. What effect do we get? In this case it is useful to consider the geometric interpretation.

When p = q = 0 and s = 1, the homogeneous coordinate of the transformed vectors a always equal to 1. Geometrically, this result is interpreted as restricting the physical transformation of the plane h = 1.

To illustrate the effect of conversion for p and q, different from zero, consider the following expression:

(1.55)

The transformed coordinate vector expressed in homogeneous coordinates, is now in a three-dimensional space defined as h = px + qy + l. This transformation is shown in Fig. 1.6, where the line AB, owned by the physical plane h = 1, is converted to a CD with a value of h <> 1, i.e., pX + qY - H + 1 = 0.

However, the results are of interest belonging to the physical plane with h = 1, which can be obtained by geometric projection line CD from the plane h <> 1 back on the plane h = 1, using for the projecting rays passing through the origin. Using a rule for similar triangles (see Fig. 1.6) we obtain:

(1.56)

or in homogeneous coordinates:

(1.57)

Fig.1.6. Geometric interpretation of homogeneous coordinates

After normalizing the expression (1.55) by dividing into the value of homogeneous coordinates h, we obtain

(1.58)

or:

(1.59)

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