Combined transformation & Homogeneous Coordinates

A)    Homogeneous Coordinates:

The homogeneous coordinates of a non – homogeneous position vector [x y] are   [x` y` h] where x=x`/h, y=y`/h and h is any real number.

(Note that h=0 has special meaning.)

One set homogeneous coordinates is always of the form [x y 1] and all other homogeneous coordinates are of the form [hx  hy  h] where h is any real number. The example of homogeneous coordinates are [3 2], [3 2 1] or [6 4 2].

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B)     Combined transformation:

1) General pivot point rotation-

Original position of object and pivot point	Translation of object so that pivot point (x,y) is at origin	Rotation about origin	Translation of object so that the pivot point is returned to position (x,y)

With a graphics package that only provides a rotate function for revolving objects about the coordinates origin, we can generate rotations about any selected pivot point (x,y) by performing the following sequence of translation – rotate – translate operations:

i)                    Translate the object so that the pivot – point position is moved to that coordinate origin.

ii)                   Rotate the object about the coordinate origin.

iii)                 Translate the object so that the pivot point is returned to its original position.

The composition transformation matrix for the sequence is obtained with the concatenation

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2)    General Fixed – Point Scaling:

         

Original position of object & fixed point	Translate object so that fixed point (x,y) is at origin	Scale object with respect to origin	Translate object so that the fixed point is returned to position (x,y)

Above figure describe a transformation sequence to produce scaling with respect to a selected fixed position (x,y) using a scaling function that can only scale relative to the coordinate origin.

i)                    Translate object so that the fixed point coincides with the coordinate origin.

ii)                   Scale the object with respect to the coordinate origin.

iii)                 Use the inverse translation of step (i) to return the object to its original position.

Concatenation the matrices for these operations produces the required scaling matrix