Download 3D Transformation notes and more Lecture notes Computer Graphics in PDF only on Docsity!

3D TRANSFORMATION

By:

Arvind Kumar

Assistant Professor

(CSE Department)

Contents

3D

Translation Rotation Scaling Reflection Shearing

3D Scaling

3D

Uniform Scaling

x '= x  sx , y '= y  sy , z '= z  sz



 

 

 

 

 

 

 





 

 

 



0 0 0 1 1

0 0 0

0 0 0

0 0 0

1

'

'

' z

y

x s

s

s z

y

x z

y

x

z^ x

y

3D Composite Scaling

3D

Scaling with a Fixed Point



 

 

 

 

 

 

 

 −− − 

 

 

 

 

 

 

 

 = 

 

 

   − − − = 00 00 01 1 1

10 10 00 00 00 0 10

0 0 00 00 00 00 01 1

01 10 00 1 '

T ( xf , yf , zf ) S ( sx , sy , sz ) T ( xf , yf , zf ) zyx '' zyxfff sx sy sz zyx fff x zy

z

y y y (^) y

Original position Inverse Translate

x (^) z x (^) z x (^) z x Translate Scaling

3D Rotation Coordinate axis

3D

About X-Axis 3D

0 sin cos 0

0 cos sin 0

z

y

x

z

y

x

z

y

x

3D Rotation Coordinate axis

3D

About Y-Axis 3D

z

y

 x 

 

 

 

 

 

 

 

  = −

 

 

 



0 0 0 1 1

sin 0 cos 0

0 1 0 0

cos 0 sin 0

1

'

'

' z

y

x z

y

x  

 

3D Rotation about an parallel axis

3D

Steps to be follows:

• Translate coincides with the the object parallel so (^) coordinatethat the rotation axis axis
• – RotationTranslate about the object that axis so .that the rotation axis is moved back to its original position x

Rotation about an arbitrary Axis

3D

1. Translate^ Steps to be Follows (x 1 , y 1 , z 1 ) to the

**2. originRotate (x’ 2 , y’ 2 , z’ 2 ) on to the z

3. axisRotate the object around the z-
4. axisRotate the axis to the original
5. orientationTranslate the rotation axis to** the original position

R-^1 T-^1

R

T

R ( ) = T −^1 R^ − x^1 ( ) R − y^1 ( ) R (^) z ( ) R (^) y ( ) R (^) x (  ) T

R

(x 2 ,y 2 ,z 2 ) (x 1 ,y 1 ,z 1 ) x z

y

Rotation about an arbitrary Axis

3D

❖ Step 2. Rotatation about x-axis

( )  

 

 

= − 

 

 

 

= − 00 0 / 0 / 10

01 0 /^0 / 00 00 sin 0 cos 0 10

R x ^01 cos^0  sin^0 ^00 bc dd cb dd

(0,b,c)

Projected Point   d

b c c dc

b b c b

2 2

2 2

cos

y sin

z

(a,b,c)  x Rotated Point

Rotation about an arbitrary Axis

3D

• Step 3. Rotate about y axis 2 22 22 2 2 2 sin , cos ld ab b c c a d

al dl == + + + = +

( )  

 

  =^ − 

 

 

  =^ − 0 / 00 0 / 10

0 / 10 0 / 00 sin 0 00 cos 0 10 R y  cos^0 ^10 sin^0 ^00 da ll da ll

(a,b,c)  d l x

y Projected Point z Rotated Point ^ (a,0,d)

Rotation about an arbitrary Axis

3D

❖ place the axis back in its initial positionStep 5. Apply the reverse transformation to ( ) ( )



 

 

 

 −



 

 

 

  −

 

 

 

 −− − − − − =

sin 0 00 cos 0 10

cos 0 10 sin 0 00 00 sin 0 cos 0 10

01 cos^0 sin^000 00 00 01 1

1 1 1 01 10 00 111

  zy  

x T R x R y

R ( ) = T −^1 R^ − x^1 ( ) R − y^1 ( ) R (^) z ( ) R (^) y ( ) R (^) x (  ) T

y l l z

x

Numerical

3D

Q1. Find an axis defined by its endpoints A(2,1,0) and B(3,3,1). the new coordinates of a unit cube 90º-rotated about

Solution:

3D

Step2. angle  Rotate axis, until it lies on the A’B’ about the x axis by and xz plane

( ) 

 

 

 

 −

0 0 05 10 0 255 55 0 R x ^015052050

1 2 1 6

cos^1555

sin 22 1 25 255 2 2 2

2 2 = + + =

= =

= + = =

l x z

y

l

B’(1,2,1) 

Projected point (0,2,1)

B”(1,0,  5 )

Solution:

3D

Step3. coincides with the z axis. Rotate axis A’B’’ about the y axis, until it

( ) 



 







 

 (^) −

0 0 06 10 606 10 030 0

630 0 66 0 R y 

cos 65 630

sin^1666 = =

= = 



x z

y

^ l A’(0,0,  6 ) B”(1,0,  5)