GEOMETRIC TRANSFORMATIONS

Ref.:

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

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 functionT 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 groupE(n ). The quotient group of E(n ) by T is isomorphic to the orthogonal groupO(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 vectorv, each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix:

In 2 dimensional space

and multiplying will give

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.

As shown below, the multiplication will give the expected result:

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.

The inverse of a translation matrix can be obtained by reversing the direction of the vector:

Similarly, the product of translation matrices is given by adding the vectors:

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.

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.

Formallya reflection is an involutiveisometry 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 spaceRn, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

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

where δij is the Kronecker delta and is given by

The formula for the reflection in the affine hyperplane is given by

Different Transformationsare summarized below.

Transformation / Example
original = ABC
image = A'B'C' / Mapping Notation
original = ABC
image = A'B'C'
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).
/ mapping: (x, y) --> (x+3, y+2)
ordered pair: [3, 2] means 3 right and 2 up


------
Additional Mapping Example
(x, y) --> (x-4, y-5)
ordered pair: [-4, -5]
means left 4 down 5 as shown by the slide arrow:
/ (x, y) --> (x+3, y+2)
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)
Reflection through the x-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 congruent).
/ mapping: (x, y) --> (x, -y)
/ (x, y) --> (x, -y)
ABC is being flipped across the x-axis (y = 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)
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 congruent).
/ 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)
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 congruent).
/ mapping: (x, y) --> (y, x)
/ (x, y) --> (y, x)
ABC is being flipped across the liney = 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)
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 congruent).
/ mapping: (x, y) --> (-y, -x)
/ (x, y) --> (-y, -x)
ABC is being flipped across the liney = -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)
Reflection through x = 1 (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 congruent).
/ mapping: (x, y) --> (2-x, y)
/ (x, y) --> (2-x, y)
ABC is being flipped across the line x = 1.
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)
Rotation about the origin
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 congruent).
/ mapping (x, y) --> (-x, -y)

turn center = origin / (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)
Rotation about the origin
quarter turn clockwise (cw) or three quarter turn counterclockwise (ccw)
  • The object and image are equidistant from a given point (turn centre).
  • The object and image are congruent (corresponding angles and sides are congruent).
/ mapping: (x, y) --> (y, -x)

turn center = origin / (x, y) --> (y, -x)
ABC is rotated 1/4 turn around the origin.
ABC is located at:
A(-3,-4), B(-2,-1), C(-5,-5)
Image A'B'C' is:
A'(-4,3),B'(-1,2),C'(-5,5)
Rotation about the origin
quarter turn counterclockwise (ccw) or three quarter turn clockwise (cw)
  • The object and image are equidistant from a given point.
  • The object and image are congruent (corresponding angles and sides are congruent).
/ mapping: (x, y) --> (-y, x)

turn center = origin / (x, y) --> (-y, x)
A(-3,-4), B(-2,-1), C(-5,-5)
ABC is rotated 1/4 turn around the origin.
ImageA'B'C' is:
A'(4,-3),B'(1,-2),C'(5,-5)
Dilatation about the origin (Enlargement or Reduction)
  • The object and image are similar when the dilatation factor  1. (corresponding angles are congruent).
  • The object and image are congruent when the dilatation factor = 1. (corresponding angles and sides are congruent).
/ mapping: (x, y) --> (3x, 3y)

dilatation centre = origin / (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)

A transformation is called anisometery, ifthe preimageand 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.

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