Transformations

Transformations

1. Rotations

   • Turns a figure about a fixed point called the center of rotation.
     • Rays drawn from the center of rotation to a point and its image form the angle of rotation.
   • Rotations can be either clockwise or counterclockwise.
   • A rotation is an isometry
   • There are three coordinate rules for rotations about the origin.
     • For a rotation of 90 degrees, (a,b)→(-b,a)
     • For a rotation of 180 degrees, (a,b)→(-a,-b)
     • For a rotation of 270 degrees, (a,b)→(b,-a)

2. Translations

   • Moves every point of a figure the same distance in the same direction.
     • The new figure produced is called an image. The original figure can be called the pre-image.
   • A translation is an isometry.
   • Vectors can be used to describe a translation.
     • A vector has a vertical component and a horizontal component and can be written which are combined in the component form of a vector.

3. Reflections and Symmetry

   • Reflections use a line like a mirror to reflect an image
   • There are four coordinate rules for reflections
     • If (a,b) is reflected over the x-axis, its image is the point (a,-b).
     • If (a,b) is reflected in the y-axis, its image is the point (-a,b).
     • If (a,b) is reflected in the line y=x, its image is the point (b,a)
     • If (a,b) is reflected in the line y=-x, its image is the point (-b,-a).
   • A reflection is an isometry.
   • Figures produced by reflections have line symmetry.
     • The line of reflection is the line of symmetry.
   • Figures have rotational symmetry if it can be mapped onto itself by a rotation of 180 degrees or less about the center of the figure.
     • The point in the center is called the center of symmetry.

4. Composite Transformations

   • Two or more transformations can be combined to form a single transformation.
     • A translation followed by a reflection can be performed one after the other to produce a glide reflection.
   • The composition of two or more isometries is an isometry
   • If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.
   • If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P.