Important Points about the Transformation for isometric projection
Note: We can also verify such Isometric transformation matrix through checking all the foreshortening factors, that is, to check where all the foreshortening factors (fx, fy, fz) are equal or not. Identify the points A, B and C on the coordinate axes as in Figure.
i) Take OA, whether O=(0,0,0) and A (1,0,0). Assume that O is projected to O' and A is projected to A'. Hence, by using equation, we have O'=(0,0,0) and A'=(2/3,-1/3,-1/3).
Consequently |O'A'| = √2/ 32 + (-1/3)2 + (-1/3)2 = √ 2/3 = fx
ii) Take OB, here O = (0,0,0) and B (0,1,0). Assume here O is projected to O' and B is projected to B'. Hence by using equation, we cans say O'=(0,0,0) and B'= (-1/3,2/3,-1/3).
Consequently |O'B'| =- √(1/3)2+ (2/3)2 + (-1/3)2 =√2 / 3 = fy
iii) Take OC, whether O=(0,0,0) and C(0,0,1). Assume here O is projected to O' and C is projected to C'. Hence, by using equation, we contain O'=(0,0,0) and C'=(-1/3,-1/3,2/3).
Consequently |O'C'|=√(-1/3)2 +(-1/3)2 +(2/3)2 = √2/3=fz
Hence, we have fx=fy=fz, that is true for Isometric projection.