This document discusses the transformation of vectors with respect to objects in 3D space. It introduces vectors and objects, and how they can be represented using matrices and frames. It then covers how to map vectors between rotated frames, translated frames, and through composite mappings using homogeneous transformation matrices. The key concepts are representing points and vectors in space with matrices, using frames to describe objects, and transforming vectors between frames through rotation, translation, and composite transformations.

1. TRANSFORMATION OF VECTORS
WITH RESPECT TO OBJECTS
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/

2. INTRODUCTION
 VECTORS – Something which has both magnitude
and direction in space.
 OBJECTS – Anything which occupies space, has
dimensions and has a physical existence.

3. 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

4. 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

5. FRAMES
 REPRESENTATION OF FRAMES
 VECTOR MATRIX NOTATION OF A FRAME

6. 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

7. 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

8. ROTATED FRAMES

9. TRANSLATED FRAMES

10. COMPOSITE MAPPING

11. DESCRIPTION OF OBJECTS IN SPACE

12. 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

13. TRANSFORMATION OF VECTORS
 ROTATION OF VECTORS
 TRANSLATION OF VECTORS
 COMPOSITE TRANSFORMATION

14. TRANSLATION OF VECTORS

15. ROTATION OF VECTORS

16. Pure Rotation about an Axis

17. COMPOSITE TRANSFORMATION

18. 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 



19. THANK YOU