GEOMETRIC TRANSFORMATIONS
Ref.: http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@transformations&lid=3
Basic Transformation Concepts
Objectives
After studying this lesson, you should be able to:

Identify types of transformations

Name the image and preimage of a mapping.

Recognize an isometry or congruence transformation.
Escher Tessellations
A Dutch artist, M.C. Escher (1898-1972), created art by using patterns known as
tessellations. Escher moved figures according to certain rules
1
Translations
A figure can be slid. This is called a translation. The object and image are congruent.
In Euclidean geometry, a translation is moving every point a constant distance in a specified
direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A
translation can also be interpreted as the addition of a constant vector to every point, or as
shifting the origin of the coordinate system. A translation operator is an operator such that
If v is a fixed vector, then the translation Tv will work as Tv(p) = p + v.
If T is a translation, then the image of a subset A under the function T is the translate of A by
T. The translate of A by Tv is often written A + v.
In an Euclidean space, any translation is an isometry. The set of all translations forms the
translation group T, which is isomorphic to the space itself, and a normal subgroup of
Euclidean group E(n ). The quotient group of E(n ) by T is isomorphic to the orthogonal group
O(n ):
E(n ) / T ≅ O(n ).
Matrix representation
Since a translation is an affine transformation (a linear transformation followed by a
translation between two affine spaces) but not a linear transformation, homogeneous
coordinates are normally used to represent the translation operator by a matrix and thus to
make it linear. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous
coordinates as w = (wx, wy, wz, 1).
To translate an object by a vector v, each homogeneous vector p (written in homogeneous
coordinates) would need to be multiplied by this translation matrix:
In 2 dimensional space
 1 0 vx 


Tv   0 1 v y  and multiplying will give
0 0 1 


 1 0 vx  px   px  vx 

  

Tv P   0 1 v y  p y    p y  v y   p  v
 0 0 1  1   1 

  

Example: Translating point p(3,5) by 2 units in the x and 4 units in the y direction, the
coordinates of the new point can be found by vector addition.
 3
 2
3  2 5
p   ; v     p  v  
 
5
 4
5  4 9
2
1

0
Tv  
0

0
0 0 vx 

1 0 vy 
0 1 vz 

0 0 1 
As shown below, the multiplication will give the expected result:
1

0
Tv P  
0

0
0 0 vx  px   px  vx 
  

1 0 v y  p y   p y  v y 

 pv
0 1 vz  pz   pz  vz 
  

0 0 1 
 1   1 
Example: Let the coordinates of a point in 3-D space be (3,6,4). If the point is to be translated
by 5 units in x direction, 4 units in y, and 7 units in z directions, find the new location of the
point.
 3
5
 35  8 
 
 

  
p   6  ; v   4   p  v   6  4   10 
 4
7
 4  7   11 
 
 

  
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
Tv1  T v
Similarly, the product of translation matrices is given by adding the vectors:
TuTv  Tu v
Because addition of vectors is commutative, multiplication of translation matrices is therefore
also commutative (unlike multiplication of arbitrary matrices).
Reflections
A figure can be reflected across a line like a mirror.
The original figure is the preimage. The result of the reflection is called the image.
3
This article is about reflection in geometry. For reflexivity of binary relations, see reflexive relation.
In mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its mirror
image. For example, a reflection of the small English letter p in respect to a vertical line would look
like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while
for reflections in the three-dimensional space one would use a plane for a mirror. Reflection sometimes
is considered as a special case of inversion with infinite radius of the reference circle.
Geometrically, to find the reflection of a point one drops a perpendicular from the point onto the line
(plane) used for reflection, and continues the same distance on the other side. To find the reflection of a
figure, one reflects each point in the figure.
1. A reflection done twice brings us back where we started.
2. A reflection preserves the distance between points.
3. A reflection does not move the points which are on the mirror.
4. Dimension of the mirror is by one smaller than the dimension of the space in which the
reflection takes places.
Formally a reflection is an involutive isometry of an Euclidean space. Here when points are subtracted
a vector is obtained, or a vector added to a point gives another point.
A figure which does not change upon undergoing a certain reflection is said to have reflection
symmetry.
Formulas
Given a vector a in Euclidean space Rn, the formula for the reflection in the hyperplane through the
origin, orthogonal to a, is given by
Re f a (v)  v  2
va
a
aa
where v·a denotes the dot product of v with a. Note that the second term in the above equation is just
twice the projection of v onto a. One can easily check that


Refa(v) = − v, if v is parallel to a, and
Refa(v) = v, if v is perpendicular to a.
Since these reflections are isometries of Euclidean space fixing the origin they may be represented by
orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose
entries are
4
Rij   ij  2
ai a j
a
2
where δij is the Kronecker delta and is given by
1, if i  j
 0 if i  j
 ij  
The formula for the reflection in the affine hyperplane is given by
Re f a,c (v)  v  2
va c
a
aa
Different Transformations are summarized below.
Transformation
Translation (Slide)



The object and image have
the same orientation.
Corresponding sides of the
object and image are
parallel -->>--.
The object and image are
congruent (corresponding
angles and sides are
congruent).
Example
Mapping Notation
original = ABC
image = A'B'C'
original = ABC
image = A'B'C'
mapping: (x, y) --> (x+3, y+2)
(x, y) --> (x+3, y+2)
ordered pair: [3, 2] means 3 right and 2 up
ABC is moved 3 units
right and 2 units up as
shown by the slide arrow
to the left.
ABC is located at:
A(7,4), B(5,2), C(3, 4)
Image A'B'C' is
A'(7+3,4+2)
B'(5+3,2+2)
C'(3+3,4+2)
or
A'(10,6), B'(8,4), C'(6, 6)
----------------------------------
5
Additional Mapping Example
(x, y) --> (x-4, y-5)
ordered pair: [-4, -5]
means left 4 down 5 as shown by the slide
arrow:
Reflection through the x-axis
mapping: (x, y) --> (x, -y)
(Flip)






ABC is being flipped
across the x-axis (y = 0).
The object and image are
equidistant from a line of
reflection.
Segments joining
corresponding sides of an
object and image are
perpendicular to the
reflection line.
The object and image are
congruent (corresponding
angles and sides are
congruent).
Reflection through the y-axis
(Flip)
The object and image are
equidistant from a line of
reflection.
Segments joining
corresponding sides of an
object and image are
perpendicular to the
reflection line.
The object and image are
congruent (corresponding
angles and sides are
(x, y) --> (x, -y)
ABC is located at:
A(4,6), B(4,2), C(2,2)
Image A'B'C' is
A'(4,-6), B'(4,-2), C'(2,-2)
mapping: (x, y) --> (-x, y)
(x, y) --> (-x, y)
ABC is being flipped
across the y-axis (x = 0).
ABC is located at:
A(4,6), B(4,2), C(2,2)
Image A'B'C' is
A'(-4,6), B'(-4,2), C'(-2,2)
6
congruent).
Reflection through y = x (Flip)





The object and image are
equidistant from a line of
reflection.
Segments joining
corresponding sides of an
object and image are
perpendicular to the
reflection line.
The object and image are
congruent (corresponding
angles and sides are
(x, y) --> (y, x)
ABC is being flipped
across the line y = x.
The object and image are
equidistant from a line of
reflection.
Segments joining
corresponding sides of an
object and image are
perpendicular to the
reflection line.
The object and image are
congruent (corresponding
angles and sides are
congruent).
Reflection through y = - x (Flip)

mapping: (x, y) --> (y, x)
ABC is located at:
A(4,6), B(4,2), C(2,2)
Image A'B'C' is
A'(6, 4), B'(2, 4), C'(2, 2)
mapping: (x, y) --> (-y, -x)
(x, y) --> (-y, -x)
ABC is being flipped
across the line y = -x.
ABC is located at:
A(4,6), B(4,2), C(2,2)
Image A'B'C' is
A'(-6,-4), B'(-2,-4), C'(-2,2)
7
congruent).
Reflection through x = 1 (Flip)



half turn clockwise (cw) or
half turn counterclockwise
(ccw)

The object and image are
equidistant from a given
point.
The object and image are
congruent (corresponding
angles and sides are
(x, y) --> (2-x, y)
ABC is being flipped
across the line x = 1.
The object and image are
equidistant from a line of
reflection.
Segments joining
corresponding sides of an
object and image are
perpendicular to the
reflection line.
The object and image are
congruent (corresponding
angles and sides are
congruent).
Rotation about the origin

mapping: (x, y) --> (2-x, y)
ABC is located at:
A(4,6), B(4,2), C(2,2)
ImageA'B'C' is
A'(-2,6), B'(-2,2), C'(0,2)
mapping (x, y) --> (-x, -y)
(x, y) --> (-x, -y)
ABC is rotated 1/2 turn
around the origin
(clockwise or
counterclockwse.
ABC is located at:
A(-3,-4), B(-2,-1), C(-5,5)
Image A'B'C' is:
A'(3, 4), B'(2, 1), C'(5, 5)
8
congruent).
turn center = origin
Rotation about the origin
mapping: (x, y) --> (y, -x)
ABC is rotated 1/4 turn
around the origin.
quarter turn clockwise (cw)
ABC is located at:
or
three quarter turn
counterclockwise (ccw)


(x, y) --> (y, -x)
A(-3,-4), B(-2,-1), C(-5,5)
The object and image are
equidistant from a given
point (turn centre).
The object and image are
congruent (corresponding
angles and sides are
congruent).
Image A'B'C' is:
A'(-4, 3), B'(-1, 2), C'(-5,
5)
turn center = origin
Rotation about the origin
mapping: (x, y) --> (-y, x)
A(-3,-4), B(-2,-1), C(-5,5)
quarter turn
ABC is rotated 1/4 turn
around the origin.
counterclockwise (ccw) or
three quarter turn clockwise
(cw)


Image A'B'C' is:
The object and image are
equidistant from a given
point.
The object and image are
congruent (corresponding
angles and sides are
congruent).
Dilatation about the origin
(Enlargement or Reduction)


(x, y) --> (-y, x)
The object and image are
similar when the
dilatation factor  1.
(corresponding angles are
congruent).
The object and image are
congruent when the
A'(4, -3), B'(1, -2), C'(5, 5)
turn center = origin
mapping: (x, y) --> (3x, 3y)
(x, y) --> (3x, 3y)
ABC is being enlarged
around the origin.
A(2,3), B(2,1), C(1,1)
Image A'B'C' is
A'(6, 9), B'(6, 3), C'(3, 3)
9
dilatation factor = 1.
(corresponding angles
and sides are congruent).
dilatation centre = origin
A transformation is called an isometery, if the preimage and image are congruent (meaning
the same shape and size).
A transformation is called a similarity transformation if the preimage and image are similar,
meaning they are the same shape but not necessarily the same size.
10