What is homogeneous coordinate? Discuss the composite transformation matrix for two successive translations and scaling.

In design and picture formation process, many times we may require to perform translation rotation and scaling to fit the picture components into their proper positions. In the previous section we have seen that each of the basic transformation can be expressed in the general matrix form. To produce a sequence of transformations with above equations, such as translation followed by rotation and then scaling, we must calculate the transformed coordinate’s one step at a time. First coordinates are translates then these translated coordinates are scaled and finally the scaled coordinates are rotated. But this sequence of transformation process is not efficient. A more efficient approach is to combine sequence of transformation into one transformation so that the final coordinate position is obtained directly from initial coordinate. This eliminates the calculation of intermediate coordinate values. In order to combine sequence of transformation terms in M₂. To achieve this we have to represent matrix M as 3 * 3 matrixes instead of 2 * 2 introducing an additional dummy coordinate W. Here points are specified by three numbers instead of two. This coordinate system and it allows us to express all transformation equations as matrix multiplication. For two dimensional transformation we can have =1.