1. COORDINATE
TRANSFORMATION
Vector andTensor Analysis

2. COORDINATE
TRANSFORMATION
Suppose that we have 2 coordinate systems in the
plane. The first system is located at origin O & has
coordinate axes xy. The second system is located at
origin O’& has coordinate axes x’y’. Now each point
in the plane has two coordinate descriptions: (x,y) or
(x’,y’), depending on which coordinate system is
used. The second system x’y’ arises from a
transformation applied to first system xy which is
called Coordinate transformation.

3. O
x
y
Tv
y’
O’
x’
P(x,y)
P’(x’,y’)

4. Translation
If the xy coordinate system is displaced to a
new position, where the direction & distance
of displacement is given by vector, v = txI + tyJ
the coordinates of a point in both systems are
related by the translation transformation Tv.
(x’, y’) = Tv(x, y)
where x’ = x – tx and y’ = y – ty.
The translation equation can be expressed as a
single matrix equation by using a column

5. vector to represent coordinate position &
translation vector.
P = x T = tx P’ = x’
y ty y’
or P’ = P – T
Rotation
A rotation is applied to the plane by
repositioning it along a circular path in xy
plane. The xy system is rotated θº abt origin.

6. TRANSLATION OF A
GEOMETRIC FIGURE IS A
SLIDE OF THE FIGURE IN
WHICH ALL POINTS MOVE
THE SAME DISTANCE IN THE
SAME DIRECTION.

7. HORIZONTAL-LEFT AND
RIGHT

8. VERTICAL-
UP
AND
DOWN

9. 54321-1
-1
-2
-3
-4
-5
-2-3-4-5
1
2
3
4
5
TRANSLATE THE FIGURE
HORIZONTALLY – 5
A
B
C
B
AC

10. -5
-3
-4
-2
54321-1
-1
-2-3-4-5
1
2
3
4
5
TRANSLATE
THE FIGURE
4UNITS
VERTICALLY.
AB
C D
B
C D

11. -5
-3
-4
-2
54321-1
-1
-2-3-4-5
1
3
2
4
5
Translate the figure
6 units vertically.
B
C
A
BC
A

12. -3
-4
-5
-2
54321-1
-1
-2-3-4-5
1
2
3
4
5
Translate the figure 4
units horizontally.
A
B
D C
A B
D C

13. vector to represent coordinate position &
translation vector.
P = x T = tx P’ = x’
y ty y’
or P’ = P – T
Rotation
A rotation is applied to the plane by
repositioning it along a circular path in xy
plane. The xy system is rotated θº abt origin.

14. The coordinates of a point in both systems are
related by rotation transformation Rθ.
(x’, y’) = Rθ(x,y)
where x’ = x cos θ + y sin θ
y’ = -x sin θ + y cos θ
The rotation equation in matrix form is
written as
where P’ = x’
P’ = Rθ.P
Rθ = cos θ
y’ -sin θ
sin θ P = x
cos θ y

15. A ROTATION of a geometric
figure is the t u r n of the
figure around a fixed point.

16. CLOCKWISE

17. COUNTER-CLOCKWISE

18. 90

19. 180

20. -5
-4
-3
54321-1
-1
-2
-2-3-4-5
2
3
4
5
Rotate the figure
clockwise90
around the origin.
A
B 1C
B
C
A

21. -5
-4
-3
-2
54321-1
-1
-2-3-4-5
1
2
3
4
5
A
B
CD
D
CB
A
ROTATE THE FIGURE
90 COUNTER-
CLOCKWISE
AROUND THE
ORIGIN.

22. -5
-4
-3
-2
54321-1
-1
-2-3-4-5
1
3
2
4
5
A
B C
A
B
C
Rotate the figure 180
counter-clockwise
around the origin.

23. -5
-4
-
2
-
3
54321
-1
1
2
3
5
4
Rotate the figure 180
clockwisearound the
origin.
A B
C
D
-5 -4 -3 -2 -1
C
B
D
A

24. Scaling
Suppose that a new coordinate system is
formed by leaving the origin & coordinate
axes unchanged, but introducing different
units of measurement along the x & y axes. If
the new units are obtained from the old units
by a scaling of sx along the x axis & sy along
the y axis, the coordinates in the new system
are related to coordinates in the old system
through the scaling transformation Ssx,sy.

25. WHERE X’ = (1/SX)X &
Y’ = (1/SY)Y. THE
COORDINATE SCALING
TRANSFORMATION USINGscaling factor sx = 2 and sy = ½.
1 2 1 2
1
2
2
4
P(2,1) P(1,2)

26. (x’,y’) = S (x,y)
The transformation equation in matrix form
is:
P’ = x’ P = x
y’
P’ = S.P
S = 1/sx 0
0 1/sy y

28. Reflection
If the new coordinate system is obtained by
reflecting the old system about either x or y
axis, the relationship b/w coordinate is given
by mirror transformation Mx & My.
(i) The mirror reflection transformation Mx
About the x-axis is given by
where x’ = x
P’ = Mx (P)
& y’ = - y

29. 1
-1
Along X-axis Along Y-axis
x’
P(1,1)
P’(1,-1)’
x
1
-1
P(1,1)
P’(-1,1)’
y y’y
y’
x’ x

30. P’ = x’ Mx = 1 0 P = x
y’ 0 -1 y
It can be represented in matrix form as
(ii) The mirror reflection transformation My
About y-axis is given by P’ = My (P)
where x’ = -x & y’ = y
It can be represented in matrix form as
P’ = x’ My = -1 0 P = x
y’ 0 1 y

32. INVERSE COORDINATE
TRANSFORMATION:
Each coordinate transformation has an inverse
which can be found by applying the opposite
Transformation.
Translation: Tv-1 = T-v,translation in
opposite direction
Rotation: Rθ = R-θ,rotation in opposite-1
direction
Scaling: S -1= Ssx,sy 1/sx,1/sy
Reflection: M -1= M & M -1=M
x x y y