Transformation Matrices


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Elements In A Matrix Show

Transformation Matrices


  • In previous articles we have seen that we can multiply two matrices together to produce a third matrix.
  • We have also seen that a vector is a special kind of matrix with only one row or column.
  • A transformation matrix is a matrix that can be used to multiply a vector to produce a vector of the same size.
  • By choosing the values in our matrix we can control how the vector is transformed.


  • Let's start with the 2-dimensional vector:
  • If we multiply it by a 2x2 matrix then we will get a vector which is also 2-dimensional:
  • By choosing different values for our matrix we can transform the vector in different ways.

Identity Matrix

  • If we choose an identity matrix as our transformation matrix then the output will equal the original vector:

Scaling Matrix

  • If we want to scale a vector then we can multiply each component by a number:

Shearing Matrix

  • Sometimes we want to slant our object:
  • In this example, the slant is parallel to the x axis:
    • x is offset by a function of y (the offset gets larger as y grows).
    • y remains unchanged.

Rotation Matrix

  • To rotate a point anti-clockwise around the origin we use the following equation:
  • This can be converted to a matrix transformation:
  • If we want to perform a clockwise rotation then we negate the matrix: hint:

Translation Matrix

  • Unfortunately it isn't possible define a matrix that translates a vector such as:
  • However, if we extend our vector by 1 dimension: and extend our transformation matrix by 1 row and 1 column: Then our new vector will be:
  • We can use the new e and f parameters of our matrix to translate our vector:

Affine Transformations

  • As part of our definition of a translation matrix we:
    • Extended the vector by 1 dimension
      • This converted it from a euclidean coordinate into a homogeneous coordinate.
    • Extended the transformation matrix by 1 row and 1 column
      • This converted it from a linear transformation into an affine transformation.
  • Each of our transformations can be rewritten as affine transformations:
    • Identity:
    • Scaling:
    • Shearing:
    • Rotation (anti-clockwise):
    • Translation:

Combining Transformations

  • The true power of transformation matrices comes from their ability to be conbined.
  • For example, if we want to translate and scale a vector then we can use:
  • Or more generally, two transformation matrices can be combined into one by multiplying them:

Order of Transformations

  • If we define a scaling matrix as:
  • and a translation matrix as:
  • then we can combine them in two different ways:
    • The translation first, and then the scaling:
    • The scale first, and then the translation (in scaled space):
    • The order of the translations goes from right to left:
      • if we have a transformation: T1T2T3
      • T3 would be applied first, then T2, and finally T1.

Extending to 3D

  • Extending to 3D is a simple case of defining a 4D vector:
  • And a 4D transformation matrix:
  • We can redefine each of our 2D transformations in 3D form:
    • Identity:
    • Scaling:
    • Shearing:
    • Rotation (around each axis):
    • Translation: