Derive the common transformation for parallel projection into a specified view plane, here the direction of projection d=aI+bJ+cK is along the normal N=n1I+n2J+n3K along with the reference point R0(x0,y0,z0).
Solution: The common transformation for parallel projection into the xy-plane in the direction of projection following figure (b) v = a I + bJ + ck, indicated by P par, V, N, Ro, contains the subsequent steps:
a) Translate the view reference point Ro of the view plane to the origin, through T-Ro
b) Perform an alignment transformation An hence that the view normal vector N of the view points in the direction K of the normal to the xy-plane. The V projection vector's direction is transformed to new vector V' = AnV.
c) Project into the xy-plane using P par, v'
d) Align k back to N, utilizing An.
e) Translate the origin back to Ro, by TRo