TRANSFORMING A POINT vis-à-vis TRANSFORMING THE AXIS

Graphics applications often require transformation of objects description form one coordinate system to another. The process of transformation involves steps like translation and rotation. This transformation can be done in two different ways either by transforming a point of the object or by transforming the axis to a point on object. Both of these methods are literally as well as mathematically opposite to each other. For example the matrix for translating point P(x,y,z) to origin the matrix would be exactly the mathematical inverse of the matrix for translating Origin to point P.

Translation Matrix to translate point P to Origin

Translation Matrix to translate Origin to Point P

This is true for rotation matrix also.