Homogeneous Coordinates

The previous sections discussed a number of transformations performed by (2×2) translation matrix. Among them are the following: rotation, mirror, scale, shear, and others. As mentioned earlier, the initial coordinate system is invariant to all these changes. However, there is a need to change the position of the origin, i.e., convert each point on the plane. This can be achieved by moving the point of origin or any other point on the plane.

(1.48)

Unfortunately, we cannot introduce a movement constant m and n in (2×2) transformation matrix, so it is not a space!

This difficulty can be overcome by using homogeneous coordinates. Homogeneous coordinates of the inhomogeneous coordinate vector [ x y ] are a triple [ x* y* h ], where x = x* / h, y = y* / h, and h - a real number. Note that the case h = 0 is special. There is always the same set of homogeneous coordinates of the form [ x y 1 ]. We chose this form to submit to the coordinate vector [ x y ] on the physical plane XY. All other homogeneous coordinates are represented as [ hx hy h ]. These coordinates do not preserve uniqueness, for example, all of the following coordinates [6 4 2], [12 8 4], [3 2 1] represents the same physical point (3 2).

The transformation matrix for homogeneous coordinates has size 3×3. In particular, the transformation matrix

(1.49)

where the action of the elements a, b, c and d of the upper part of the (2×2) matrix, corresponds exactly to the actions discussed above. Elements m and n are the coefficients of translation in x and y directions, respectively. Complete two-dimensional transformation matrix has the form

(1.50)

Note that each point of the plane and even the origin x = y = 0 now can be converted.

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