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TRANSFORMATION OF VECTORS
WITH RESPECT TO OBJECTS
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/
INTRODUCTION
 VECTORS – Something which has both magnitude
and direction in space.
 OBJECTS – Anything which occupies space, has
dimensions and has a physical existence.
Matrix Representation
- Representation Of A Point In Space
A point P in space :
3 coordinates relative to a reference frame
Representation of a point in space
^ ^ ^
P  ax iby jcz k
Representation of a vector in space
-Representation of a Vector in Space
A Vector P in space :
3 coordinates of its tail and of its head
^ ^ ^
P  ax iby jcz k
 x
 z 
 

 y
w
P  
MATRIX REPRESENTATION
Where w is Scale
factor
FRAMES
 REPRESENTATION OF FRAMES
 VECTOR MATRIX NOTATION OF A FRAME
Representation of a frame in a frame
Each Unit Vector is mutually perpendicular. :
normal, orientation, approach vector

nz oz az Pz 
0 0 0 1
y y y y
P
nx ox ax Px
n

F  o a
Representation of a Frame in a Fixed
Reference Frame
MAPPING OF FRAMES
 MAPPING – Changing the description of a point in
space from one frame to another.
 MAPPING BETWEEN ROTATED FRAMES
 MAPPING BETWEEN TRANSLATED FRAMES
 COMPOSITE MAPPING
ROTATED FRAMES
TRANSLATED FRAMES
COMPOSITE MAPPING
DESCRIPTION OF OBJECTS IN SPACE
Representation of an object in space
An object can be represented in space by attaching a frame
to it and representing the frame in space.

nx ox ax Px 

 ny oy ay Py

nz oz az Pz 
 0 0 0 1 


Fobject
Representation of a Rigid Body
TRANSFORMATION OF VECTORS
 ROTATION OF VECTORS
 TRANSLATION OF VECTORS
 COMPOSITE TRANSFORMATION
TRANSLATION OF VECTORS
ROTATION OF VECTORS
Pure Rotation about an Axis
COMPOSITE TRANSFORMATION
Homogeneous Transformation Matrices
A transformation matrices must be in square form.
•It is much easier to calculate the inverse of square matrices.
•To multiply two matrices, their dimensions must match.

nx ox ax Px 

F  n y o y a y Py

nz oz az Pz 
 0 0 0 1 


THANK YOU

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