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Computer

Graphics

Lecture 18

3-D

Transformations-II

Normalization

Note that the process of

moving the points so that

the POV is at the origin

looking down the +Z axis

is called normalization.

For Rotation

Rotating a point requires

that we know:

• 1. the coordinates for the

point, and

• 2. the rotation angles

Rotating a Point

Repositions an object along a

circular path in xy-plane

Rotation Angle θ

Rotation Point (x (^) r , y (^) r )

 θ is +ve  counterclockwise

rotation

 θ is -ve  clockwise rotation

 θ is zero ?

Replacing r cos Ф with x and

r sin Ф with y, we have:

x ′ = x cos θ – y sin θ

and

y ′ = x sin θ + y cos θ

Column vector representation:

P′ = R. P

Where

 

  

 −

θ θ

θ θ sin cos

cos sin R (^)  

  



y

x  P 

  



'

' ' y

x P

… Now in 3D

Rotation can be about any of

the three axes:

About z-axis (i.e. in xy plane)

About x-axis (i.e. in yz plane)

About y-axis (i.e. in xz plane)

Roll : around z-axis

Yaw: around y-axis

Rotation about z-axis

(i.e. in xy plane):

• x ′ = x cos θ – y sin θ
• y ′ = x sin θ + y cos θ
• z’ = z

Alright, but what about

rotations w.r.t. other

axes?

Cyclic Permutations of Coordinate Axes

Cyclic permutation