Computer Graphic - Transformations in 3d

2.  Right handed space consistent with math.
 Left handed space suitable to screens.
 To transform from right to left negate the z values.

8.  There are five steps for reflection about an arbitrary
point in space:
1. Translate to the origin by –(x0,y0,z0).
2. Rotate (x’,y’,z’) about x-axis.
3. Rotate (x’,y’,z’) about y-axis.
4. Rotate about z-axis.
5. Inverse step 2 By changing the sin sign.
6. Inverse step 3 By changing the sin sign.
7. Inverse step 1 By changing the tx, ty and tz sign.

9.  EX: Find the new coordinate of a unit cube 90 degree
rotated about an axis defined by its end points A(2,1,0)
and B(3,3,1).

10.  Step 1: Translate to origin
 a= 3-2= 1
 b= 3-1= 2
 c= 1-0= 1
1 0 0 -2
0 1 0 -1
0 0 1 0
0 0 0 1

11.  Step 2: Rotation about x-axis
 D= 𝑏2 + 𝑐2 = 22 + 12 = 5
 Sin= b/d= 2/ 5
 Cos= c/d= 1/ 5
1 0 0
0
0 1/ 5 -2/ 5
1
0 2/ 5 1/ 5
1
0 0 0
1

12.  Step 3: Rotation about y-axis
 L= 𝑎2 + 𝑏2 + 𝑐2 = 6
 Sin= a/L= 1/ 6
 Cos= d/L= 5/ 6
5/ 6 0 -1/ 6 0
0 1 0 0
1/ 6 0 5/ 6 0
0 0 0 1

13.  Step 4: Rotation about z-axis by 90 degree
0 -1 0 0
1 0 0 0
0 0 1 0
0 0 0 1

14.  Step 5: Inverse Rotation about x-axis
1 0 0
0
0 1/ 5 2/ 5
1
0 -2/ 5 1/ 5
1
0 0 0
1

15.  Step 6: Inverse Rotation about y-axis
5/ 6 0 1/ 6 0
0 1 0 0
−1/ 6 0 5/ 6 0
0 0 0 1

16.  Step 7: Inverse of Translation
 Then we multiply all the matrices with each other.
1 0 0 2
0 1 0 1
0 0 1 0
0 0 0 1

17.  EX: Find the rotation for the point (1,2,1) by 90 degree
(1,2,3)
(4,6,7)

18.  Step 1: Translate to origin
 A= 4-1= 3
 B= 6-2= 4
 C= 7-3= 4
1 0 0 -1
0 1 0 -2
0 0 1 -3
0 0 0 1

19.  Step 2: Rotation about x-axis
 D= 𝑏2 + 𝑐2 = 42 + 42 = 32
 Sin= b/d= 4/ 32
 Cos= c/d= 4/ 321 0 0
0
0 4/ 32 -4/ 32 1
0 4/ 32 4/ 32
1
0 0 0
1

20.  Step 3: Rotation about y-axis
 L= 𝑎2 + 𝑏2 + 𝑐2 = 41
 Sin= a/L= 3/ 41
 Cos= d/L= 32/ 6
32/ 41 0 -3/41 0
0 1 0 0
3/ 41 0 32/ 41 0
0 0 0 1

21.  Step 4: Rotation about z-axis by 90 degree
0 -1 0 0
1 0 0 0
0 0 1 0
0 0 0 1

22.  Step 5: Inverse step 2
1 0 0
0
0 4/ 32 4/ 32 1
0 -4/ 32 4/ 32 1
0 0 0
1

23.  Step 6: Inverse step 3
32/ 41 0 3/ 41 0
0 1 0 0
-3/ 41 0 32/ 41 0
0 0 0 1

24.  Step 1: Inverse step 1
 Then we multiply all the matrices with each other and
with the point (1,2,1).
1 0 0 1
0 1 0 2
0 0 1 3
0 0 0 1