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Abstract

Transforms are used to scale, translate, rotate, reflect and shear shapes and objects. For example, if we start with a 2D point P(x, y), it can be transformed into P′(x′, y′) by manipulating the original coordinates x and y using
$$\begin{array}{rcl} x'& =& ax + by + e \\ y'& =& cx + dy + f\end{array}$$
Similarly, a 3D point P(x, y, z) can be transformed into P′(x′, y′, z′) using
$$\begin{array}{rcl} x'& =& ax + by + cz + k \\ y'& =& dx + ey + fz + l \\ z'& =& gx + hy + jz + m\end{array}$$
By choosing different values for a, b, c, . . . . we can translate, shear, scale, reflect or rotate a shape.

Although algebra is the basic notation for transforms, it is also possible to express them as matrices, which provide certain advantages for viewing the transform and for interfacing to various types of computer graphics hardware. We begin with an algebraic approach and then introduce matrix notation.

Keywords

Direction Cosine Matrix Notation Axial System Perspective Projection Unit Quaternion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  • John Vince
    • 1
  1. 1.RingwoodUnited Kingdom

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