5.3 Isometries & Dilations

 

Isometry:

In Geometry an Isometry is a transformation of a figure by either a Translation, Reflection, Rotation or a combination of  Translations/Reflections/Rotations.

Translation:

A Translation is an Isometry, meaning the Preimage ( original image ) and its translated Image ( what the preimage was changed in to ) are the same shape and size.

A translation ( slide or glide ) shifts every point of a figure the same distance in the same direction. A figure that is transformed by a translation remains congruent to its preimage. Its side lengths, angle measures, and other properties remain the same. Translation changes nothing but the location of a figure.

Example A: Translation

A Translation can be identified by a vector, which is a quantity that has both magnitude and direction.

The direction of a vector is the orientation of the vector, which is determined by the angle or the slope the vector makes with a horizontal line going through a point on the preimage to a corresponding point on the image. 

      

The magnitude of a vector is the length of a vector, from a point on the preimage  to a corresponding point on the image. 

 

  
         
 

Rotation:

A Rotation is an Isometry, meaning the Preimage ( original image ) and its translated Image ( what the preimage was changed in to ) are the same shape and size.

A rotation ( turn ) is a transformation about a point. A figure that is transformed by a rotation remains congruent to its preimage. Its side lengths, angle measures, and other properties remain the same. Rotation changes nothing but the direction of a figure or location and the direction of a figure, depending upon the point the figure is rotated about.

Example B: Rotation

The Center of Rotation is the point that the figure is rotated about.
There are 3-possible locations of the Center of Rotation in relation to the figure:

     1.   The Center of Rotation is inside of the figure
           ( think of a wheel on an axle )

     2.   The Center of Rotation is a vertex of the figure
           ( think of the hinge on your Chromebook )

     3.   The Center of Rotation is outside of the figure
           ( think of a child on a swing )

The direction of the rotation can be identified by a Degree Measure.  A Positive Degree Measure would indicate a Counter Clockwise Rotation of the identified number of degrees and a Negative Degree Measure would would indicate a Clockwise Rotation of the identified number of degrees.

  
             
 

Reflection:

A Reflection is an Isometry,  meaning the Preimage ( original image ) and its translated Image ( what the preimage was changed in to ) are the same shape and size.

A reflection ( mirror image ) is a transformation that reflects every point in a figure over a given line. Reflection changes nothing but the orientation of a figure and the location of a figure, depending upon the reflection line.

Example C: Reflection

The Reflection Line can be thought of in 2-different ways

     1. The Reflection Line is the Perpendicular Bisector of a
        segment from a point on the Preimage to the corresponding
        point on the Image.

         Diagonal Reflection Line:  Identify the preimage coordinate ( x , y )
        then identify the coordinate of where you want the image to go.
        Find the slope and midpoint, then draw a perpendicular line
        through the midpoint.

         Horizontal or Vertical Reflection Line:  Identify the distance
        ( magnitude )  from the preimage to the image you want.  Draw a
        perpendicular line, half of the distance from the preimage to the image,
        to the intended reflection between the preimage and image.

   2. The Reflection Line is one of the sides of the Preimage.
       The Image and Preimage will share a common side.

 

  

Note:  Due to limitations of Scratch, we are going to focus on Horizontal Reflections only.
 

Composition of Isometries:

The Composition of two, or more,  Isometries is an Isometry.  Recall that translations, rotations, and reflections are all isometries. A composite transformation with two or more of these transformations will also be an isometry.

Example D: Composition of Isometries

While there are many different ways to mix isometries together, we are going to focus on two.

A Glide Reflection is a composition of a translation and a reflection across
a line parallel to the translation vector. Since a glide reflection combines
translation and reflection, it is an isometry.

A Tessellation is a repeating pattern of plane figures that completely covers a plane with no gaps or overlaps. The simplest kind of tessellation is a regular tessellation: a repeating pattern of congruent regular polygons.
Glide Reflection
 Tessallation

Dilations:

A Dilation maps a figure ( Preimage ) to a similar figure ( Image ).  A dilation is a Transformation that changes the size of a figure but not its shape. The multiplier used on each dimension of a figure to change it into a similar figure is the Scale Factor ( Ratio of Similitude ).  A dilation that results in an image smaller than its preimage is called a Reduction or a Contraction. A dilation that results in an image larger than its preimage is called an Enlargement or an Expansion.  Dilations require a center and a scale factor. The Center of Dilation is the intersection of lines that connect each point of the image with the corresponding point of the preimage.  A dilation with a Scale Factor > 1 will result in an image larger than the preimage and farther away from the center of dilation.  A dilation with a 0 < Scale Factor < 1 will result in an image smaller than the preimage and between the preimage and the center of dilation.

Example E: Dilation

Dilations of 2-dimension figures can be used to give the impression of a 3-dimension object or impression of figures being closer or further away from the observer.

Here we have 2-dimensional figure ( triangle ) being reduced and moving toward the center of dilation.  Giving the observer the impression of the figure shrinking ( Contraction ) from view.

Then, the 2-dimensional figure appears to move away from the center of dilation and enlarges ( Expansion ) as it appears to move toward the observer.

  
Contraction
Expansion

 

Scratch

Scratch is the world’s largest coding community for children and a coding language with a simple visual interface that allows young people to create digital stories, games, and animations. Scratch is designed, developed, and moderated by the Scratch Foundation, a nonprofit organization. Scratch promotes computational thinking and problem solving skills; creative teaching and learning; self-expression and collaboration; and equity in computing. Scratch is always free and is available in more than 70 languages.

1.  Click the link to Scratch -  https://scratch.mit.edu

2.  Click on ,you will need to create your own Username & Password ( write it down! ),
     answer the given questions and use your school email account.

3.  After you "Sign In", click on to open your "Stuff"

4.  After you get to your "Stuff", click on to open the editor.

 

Practice Problems

Take a screenshot of your block code & isomerty then paste it into your jamboard

A.  Create a Translation of Triangle40 to do the following:

     From ( 0 , 0 ), clone the triangle and translate it...
  • Magnitude = 100, Direction = Horizontal & Right
     
  • Magnitude = 125, Direction = Vertical & Down
     
  • Magnitude = 75, Direction =   Slope of 2 / 3
     
  • Magnitude = 150, Direction = Slope of - 3 / 4
 		 Notes:
  • Create a called "5.3ppA"
     
  • Download the "Triangle40" sprite, then import it into Scratch
  • Duplicate the "Triangle40" sprite, and then change the name
    of the copy to "TranslationTriangle40" before you begin.

  • Set coordinates to ( 0 , 0 )
     
  • So you have time to get a screenshot, set the "wait seconds" block as follows:

     
B.  Create a Rotation of Square40 to do the following:
  • Center of Rotation = Inside, Clockwise 120°
     
  • Center of Rotation = Vertex, Rotation Degree Measure = 120°
      
  • Center of Rotation = Outside, Counter Clockwise 150°
          
  • Center of Rotation = Outside, Rotation Degree Measure = -150°
 Notes:
  • Create a called "5.3ppB"
     
  • Download the "Square40" sprite, then import it into Scratch
  • Duplicate the "Square40" sprite 4-times,
    then change the name & the center of the sprite as follows:
    -  "RotationCenter", Center = Intersection of the Diagonals
    -  "RotationVertex", Center = Any Vertex you choose
    -  "RotationOutsideCCW", Center = Dot any where above the Square
    -  "RotationOutsideCW", Center = Dot any where below the Square

  • Set coordinates to ( 0 , 0 ) & Direction = 90
     
  • Use the "ghost effect" block to fade out the original square
    0% fade,
    75% fade
C.  Create a Horizontal Reflection of Flag100 to do the following:
  • From ( -100 , 0 ) To ( 100 , 0 )
     
  • From ( 150 , -50 ) To ( -150 , -50 )
     

     		Notes:
  • Create a called "5.3ppC"
  • Click: to open the extension & add:
  • Download the "Flag100" & "Pencil" sprites, then import it into Scratch
  • Duplicate the "Flag100" sprite twice,
    then change the name of the sprite as follows:
    -  "RotationCenter", Center = Intersection of the Diagonals
    -  "RotationVertex", Center = Any Vertex you choose
  • Set coordinates to ( 150 , 0 )
     
  • These blocks will Flip the Flag
  • These blocks will tell the Pencil where to draw the Reflection Line.
D.  Create a Glide Reflection of Trangle40 to do the following:
  • Translation: Vertically 50 Pixels from ( 50 , -150 )
  • Reflection:  Over the Y-Axis of 50 pixels
     

Notes:

  • Create a called "5.3ppD"
     
  • Import the "Trangle40 " sprite into Scratch
  • Duplicate the "Trangle40 " sprite, and then change the name
    of the copy to "TrangleColor" before you begin.
     
  • Use to color each section of the triangle a different color
        
  • Create a "New Variable" called "Step"
     
  • The Reflection Line can be thought of as an imaginary line, so we do
    not need to actually draw it out if we don't want to..
E.  Create a Tessellation of Square40 to do the following:
  • Translation: Vertically -40 Pixels from ( 0 , -150 ) for 8-rows
       
  • Rotation::  180°
      
  • Translation: Horizontally -40 Pixels for 4 columns
     

Notes

  • Create a called "5.3ppE"
     
  • Import the "Square40 " sprite into Scratch
  • Duplicate the "Square40 " sprite, and then change the name
    of the copy to "SquareColor" before you begin.
        
  • Create a "New Variable" called "Column", "Row", & "Move"
       
  • Use to color each section of the square a different color
F.  Create a Dilation of Tree100 to do the following:
  • Contraction Dilation of 20- trees from ( -250 , 0 ) to ( -3 , 171 )
       
  • Expansion Dilation of 20- trees from ( 3 , 171 ) to ( 250 , 0 )
      
  • Maybe even a road...

Notes

  • Create a called "5.3ppF"
     
  • Import the "Tree100" & "Road" sprites into Scratch
  • Duplicate the "Tree100" sprite, and then change the name
    of the copy to "Trees" before you begin.
        
  • Create a "New Variable" called "ChangeX", "ChangeY",
    "Image"    , & "Size".   You should see the variable "X".
       

Assignment:

Using what you have learned in this lesson, Create the following Isometries & Dilations

A.  Create a Translation of Triangle40 to do the following:

B.  Create a Rotation of Square40 to do the following:

C.  Create a Horizontal Reflection of Flag100 to do the following:

D.  Create a Glide Reflection of Trangle40 to do the following:

E.  Create a Tessellation of Square40 to do the following:

F.  Create a Dilation of Tree100 to do the following:		 Notes:
  • Create a called "5.3ppA"
     
  • Download the "Triangle40" sprite, then import it into Scratch
  • Duplicate the "Triangle40" sprite, and then change the name
    of the copy to "TranslationTriangle40" before you begin.

  • Set coordinates to ( 0 , 0 )
     
  • So you have time to get a screenshot, set the "wait seconds" block as follows: