This document discusses different types of coordinate transformations, including translation, rotation, scaling, and reflection. Translation moves all points the same distance in the same direction. Rotation turns the coordinate system around a fixed point. Scaling changes the units of measurement along the axes. Reflection mirrors the coordinate system across an axis. Each transformation has a corresponding inverse that undoes the original transformation.

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