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Raster Affine Transformation

This page explains the basic concepts of Affine Transformation in Sedona Raster.

Affine Transformations

Affine transformations are a fundamental concept in computer graphics, geometry, and image processing that involve manipulating an object in a way that preserves lines and parallelism (but not necessarily distances and angles). These transformations are linear transformations followed by a translation, which means they can translate, scale, rotate, and shear objects without altering the relative arrangement of points, lines, or planes.

Components of Affine Transformations

Affine transformations can be represented as a matrix operation. In two-dimensional space, a typical affine transformation matrix is a 3x3 matrix as follows:

| ScaleX  SkewX   TranslationX |
| SkewY   ScaleY  TranslationY |
| 0       0       1            |

Here, ScaleX, ScaleY, SkewX, SkewY, TranslationX, and TranslationY are parameters that define the transformation:

  • ScaleX and ScaleY are scaling factors for the x and y axes, respectively.
  • SkewX and SkewY introduce shearing and are responsible for "skewing" the shape.
  • TranslationX and TranslationY are translation parameters that move the shape in the x and y directions, respectively.

Types of Affine Transformations

  1. Translation: Moves every point of a figure or space by the same distance in a given direction. This primarily affects the TranslationX and TranslationY components.

  2. Scaling: Multiplies the coordinates of each point by a constant (ScaleX for x-axis and ScaleY for y-axis), enlarging or reducing its size. Scaling can be uniform (the same factor for both axes) or non-uniform (different factors for each axis).

  3. Rotation: Rotates the object about a point (usually the origin or a specified point). This can be expressed through combinations of ScaleX, ScaleY, SkewX, and SkewY where these parameters are derived from the cosine and sine of the rotation angle.

  4. Shearing: Transforms parallel lines to still be parallel but moves them so that they are no longer perpendicular to their original orientations. This affects the SkewX and SkewY components.

  5. Reflection: Flips the object over a specified axis, which can be achieved by combining scaling and rotation.

Mathematical Properties

  • Collinearity and Concurrency: Affine transformations preserve points on a line (collinearity) and the intersection of lines (concurrency).
  • Ratios of Segments: They also preserve the ratios of distances between points lying on a straight line.

Components of Affine Transformations

In affine transformations, which are integral to manipulating graphics, images, and geometric data, the terms ScaleX, ScaleY, SkewX, and SkewY refer to specific types of transformations that alter the shape and position of objects:

ScaleX and ScaleY

  • ScaleX: This parameter represents the scaling factor along the x-axis. It modifies the width of an image or object. Values greater than 1 increase the width, values less than 1 decrease it, and negative values reflect the object along the x-axis while scaling.

  • ScaleY: This parameter represents the scaling factor along the y-axis. It affects the height of the object. Similarly to ScaleX, values greater than 1 enlarge the object vertically, values less than 1 reduce it, and negative values invert it along the y-axis.

SkewX and SkewY

  • SkewX: This parameter is used to skew or shear the object along the x-axis. It shifts each point's x-coordinate in proportion to its y-coordinate, creating a slanting effect. This transformation is useful for creating the illusion of depth or perspective in 2D representations.

  • SkewY: Corresponding to SkewX, SkewY skews the object along the y-axis. It alters each point's y-coordinate relative to its x-coordinate, which also creates a slanting effect, but in the vertical direction.

These transformations are typically used together in a transformation matrix, which allows them to be applied to objects in a combined and coherent way. Here's a typical representation of such a matrix:

| ScaleX  SkewX   TranslationX |
| SkewY   ScaleY  TranslationY |
| 0       0       1            |

These parameters can be combined in various ways to perform complex transformations such as rotations, translations, scaling, and shearing of images or shapes in both 2D and 3D graphics applications.

Last update: May 15, 2024 03:31:55