transformation.

Transformation
?Transformation is the backbone of computer graphics, it enables us to manipulate the shape, size, and location of the object. It can be used to effect the following changes in a geometric object:
?Change the location
?Change the shape
?Change the size
?Rotate
?Copy
?Generate a surface from a line
?Generate a solid from a surface
?Animate the object
Basic transformations are translation, scaling, rotation, reflection and, shearing. We look at transformations as ways of moving the points that describe one or more geometric objects to new locations. Although there are many transformations that will move a particular point to a new location, there will almost always be only a single way to transform a collection of points to new locations while preserving the spatial relationships among them. Hence, although we can find many matrices that will move one corner of our color cube from P0 to Q0, only one of them, when applied to all the vertices of the cube, will result in a displaced cube of the same size and orientation.
Translation
?Translation is an operation that displaces points by a fixed distance in a given direction. To specify a translation we need only to specify a displacement vector d, because the transformed points are given by:
p2=p1+d
?This equation can be written is a scalar form as:
x2=x1+dx and y2=y1+dy
?In matrix form, the translation is expressed as
The translation matrix is given by:
?This method of representing translation using the addition of column matrices does not combine well with our representations of other affine transformations. However, we can also get this result using the matrix multiplication:
p = TP, where
And for 3D transformation we have:
, ??=????????
We can obtain the inverse of a translation matrix either by applying an inversion algorithm or by noting that if we displace a point by the vector d, we can return to the original position by a displacement of( - d )