Transformations in Geometry to Know for Elementary Algebraic Geometry

Transformations in geometry are essential for understanding how shapes move and change. They include translations, rotations, reflections, and more, each preserving or altering properties like size and shape, which connects deeply to concepts in Elementary Algebraic Geometry.

1. Translation

   • Moves every point of a shape or object the same distance in a specified direction.
   • Can be represented mathematically by adding a vector to each point's coordinates.
   • Preserves the shape and size of the object, making it a type of isometry.

2. Rotation

   • Involves turning a shape around a fixed point, known as the center of rotation.
   • The angle of rotation determines how far the shape is turned, measured in degrees or radians.
   • Like translation, rotation preserves the shape and size of the object.

3. Reflection

   • Flips a shape over a line (the line of reflection), creating a mirror image.
   • Each point on the original shape is equidistant from the line of reflection as its corresponding point on the image.
   • Maintains the shape and size, thus classified as an isometry.

4. Dilation (Scaling)

   • Changes the size of a shape while maintaining its proportions, either enlarging or reducing it.
   • Defined by a center of dilation and a scale factor that determines how much the shape is scaled.
   • Alters the size but not the shape, distinguishing it from isometries.

5. Shear

   • Distorts the shape by shifting one part of it in a specific direction, while keeping the other part fixed.
   • Can be horizontal or vertical, depending on the direction of the shift.
   • Changes the shape but not the area, making it a type of affine transformation.

6. Glide reflection

   • Combines a translation with a reflection across a line parallel to the direction of the translation.
   • Each point is first translated and then reflected, resulting in a unique transformation.
   • Preserves distances and angles, thus classified as an isometry.

7. Isometry

   • A transformation that preserves distances and angles between points.
   • Includes translations, rotations, reflections, and glide reflections.
   • Essential for studying congruence in geometry, as the shape and size remain unchanged.

8. Similarity transformation

   • A transformation that alters the size of a shape but maintains its proportions and angles.
   • Includes dilations and isometries, allowing for the comparison of shapes that are similar but not necessarily congruent.
   • Important for understanding geometric relationships and properties.

9. Affine transformation

   • A transformation that preserves points, straight lines, and planes, but not necessarily distances and angles.
   • Includes translations, rotations, scalings, and shears.
   • Useful in computer graphics and image processing for manipulating shapes and images.

10. Projective transformation

    • A transformation that maps points in a projective space, allowing for perspective changes.
    • Can alter parallel lines to meet at a point, changing the nature of shapes and their relationships.
    • Important in advanced geometry and applications such as computer vision and graphics.