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3 Dtransformation

The document provides an overview of 3D transformations in computer graphics, detailing various types such as translation, rotation, scaling, reflection, and shearing. It explains the mathematical representations and equations used for each transformation type, emphasizing their importance in positioning and modeling objects in a 3D space. The document also discusses the use of homogeneous coordinates and transformation matrices in performing these transformations.

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charvi2617
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15 views22 pages

3 Dtransformation

The document provides an overview of 3D transformations in computer graphics, detailing various types such as translation, rotation, scaling, reflection, and shearing. It explains the mathematical representations and equations used for each transformation type, emphasizing their importance in positioning and modeling objects in a 3D space. The document also discusses the use of homogeneous coordinates and transformation matrices in performing these transformations.

Uploaded by

charvi2617
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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3D TRANSFORMATION

CONTENTS
 Transformation
 Types of transformation
 Why we use transformation
 3D Transformation
 3D Translation
 3D Rotation
 3D Scaling
 3D Reflection
 3D Shearing
TRANSFORMATION
 Transformations are a fundamental part of the computer
graphics. Transformations are the movement of the object in
Cartesian plane .
TYPES OF
TRANSFORMATION
 There are two types of transformation in computer graphics.
1) 2D transformation
2) 3D transformation
 Types of 2D and 3D transformation
1) Translation
2) Rotation
3) Scaling
4) Shearing
5) Mirror reflection
WHY WE USE
TRANSFORMATION
 Transformation are used to position objects , to shape object , to
change viewing positions , and even how something is viewed.

 In simple words transformation is used for


1) Modeling
2) viewing
3D TRANSFORMATION
 When the transformation takes place on a 3D plane .it is called
3D transformation.
 Generalize from 2D by including z coordinate
Straight forward for translation and scale, rotation more difficult

Homogeneous coordinates: 4 components a b c tx 


d e f t y 
Transformation matrices: 4×4 elements 
g h i tz 
 
0 0 0 1
3D TRANSLATION
 Moving of object is called translation.
 In 3 dimensional homogeneous coordinate representation , a
point is transformed from position P = ( x, y , z) to P’=( x’, y’,
z’)
 This can be written as:-
Using P’ = T . P
 x  1 0 0 tx   x
 y   0 1 0 t y   y 
  
 z   0 0 1 tz   z 
     
 1  0 0 0 1  1 
3D TRANSLATION
 The matrix representation is equivalent to the three equation.
x’=x+ tx , y’=y+ ty , z’=z+ tz
Where parameter tx , ty , tz are specifying translation distance for the
coordinate direction x , y , z are assigned any real value.
3D ROTATION
Where an object is to be rotated about an axis that is parallel to one
of the coordinate axis, we can obtain the desired rotation with
the following transformation sequence.
Coordinate axis rotation
Z- axis Rotation(Roll)
Y-axis Rotation(Yaw)
X-axis Rotation(Pitch)
COORDINATE AXIS
ROTATION
 Obtain rotations around other axes through cyclic permutation

x yzx
of coordinate parameters:
X-AXIS ROTATION
The equation for X-axis rotation
x’ = x
y’ = y cosθ – z sinθ
z’ = y sinθ + z cosθ

 x' 1 0 0 0  x 
 y ' 0 cos   sin  0  y 
   
 z '  0 sin  cos  0  z 
     
 1  0 0 0 1  1 
Y-AXIS ROTATION
The equation for Y-axis rotaion
x’ = x cosθ + z sinθ
y’ = y
z’ = z cosθ - x sinθ

 x'  cos  0 sin  0  x 


 y '  0 1 0 0  y 
   
 z '   sin  0 cos  0  z 
     
1  0 0 0 1  1 
Z-AXIS ROTATION

The equation for Y-axis rotaion


x’ = x cosθ – y sinθ
y’ = x sinθ + y cosθ
z’ = z

 x' cos   sin  0 0  x 


 y '  sin  cos  0 0  y 
  
 z'  0 0 1 0  z 
     
1  0 0 0 1  1 
3D SCALING
 Changes the size of the object and repositions the object relative
to the coordinate origin.

 x   s x 0 0 0  x 
 y   0 s 0   
0  y 
  y

 z   0 0 sz 0  z 
     
1 0 0 0 1  1 
3D SCALING
 The equations for scaling
x’ = x . sx
Ssx,sy,sz y’ = y . sy
z’ = z . sz
3D REFLECTION
 Reflection in computer graphics is used to emulate reflective
objects like mirrors and shiny surfaces
 Reflection may be an x-axis
y-axis , z-axis. and also in
the planes xy-plane,yz-plane , and
zx-plane.
Reflection relative to a given
Axis are equivalent to 180
Degree rotations
3D REFLECTION
 Reflection about x-axis:-
x’=x y’=-y z’=-z
1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 1
Reflection about y-axis:-
y’=y x’=-x z’=-z
3D REFLECTION
 The matrix for reflection about y-axis:-
-1 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 1
 Reflection about z-axis:-
x’=-x y’=-y z’=z

-1 0 0 0
0 -1 0 0
0 0 1 0
0 0 0 1
3D SHEARING
 Modify object shapes
 Useful for perspective projections
 When an object is viewed from different directions and at
different distances, the appearance of the object will be
different. Such view is called perspective view. Perspective
projections mimic what the human eyes see.
3D SHEARING
E.g. draw a cube (3D) on a screen (2D) Alter the values for x and y
by an amount proportional to the distance from zref
3D SHEARING
 Matrix for 3d shearing
 Where a and b can
Be assigned any real
Value.
Thank You

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