Modeling Transformations

Contents 2D-Transformations -Translation – 2D - Scaling – 2D -Rotation – 2D Mirror Reflection Inverse 2D – Transformations Homogeneous coordinates -Translations with homogeneous -Scaling with homogeneous -Rotation with homogeneous

2D-Transformations Transformations are a fundamental part of computer graphics. Transformations are used to position objects, to shape objects, to change viewing positions, and even to change how something is viewed (e.g. the type of perspective that is used). There are 4 main types of transformations that one can perform in 2 dimensions: Translations Scaling Rotation Reflection BACK

Translation – 2D Shifting of an object from one position to anther is called translation Translated coord. are , x’ = x + dx y’ = y + dy BACK

Scaling – 2D To scale an object by a vector v = (vx, vy, vz), each point p = (px, py, pz) would need to be multiplied with this scaling matrix: Types of Scaling: Differential ( sx != sy ) Uniform ( sx = sy ) BACK

Rotation – 2D rotated original Coordinates before rotation Coordinates after rotation BACK

Rotation – 2D BACK

Mirror Reflection Back

Inverse 2D - Transformations BACK

Homogeneous coordinates To perform a complex transformation, you usually make it by combining a number of basic transformations. In order to represent all transformations in the same form, computer scientists have devised what are called homogeneous coordinates. Add a third coordinate, z A 2D point is a 3 coordinates vector: BACK

Translations with homogeneous BACK

Scaling with homogeneous BACK

Rotation with homogeneous BACK

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