Geometric Transformations for Computer Graphics

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Presentation transcript:

Geometric Transformations for Computer Graphics

2D Translation

2D Rotation

2D Scaling

Homogeneous Coordinates

2D Translation 2D Rotation 2D Scaling

Inverse transformations: Composite transformations: Composite translations:

Composite Rotations: Composite Scaling:

General 2D Rotation Move to origin Rotate Move back

General 2D Scaling Move to origin Scale Move back

2D Directional Scaling

2D Reflections

1 2 3 1 2 3 1 2 3 3 1 2

Geometric Transformations by Rasterization The transformed shape needs to be filled. A whole scan-line filling is usually in order. However, simple transformations can save new filling by manipulating blocks in the frame buffer. Translation: Move block of pixels of frame buffer into new destination.

90° counterclockwise rotation Destination pixel array Rotated pixel block RGB of destination pixel can be determined by averaging rotated ones (as antialiasing)

Very similar to 2D. Using 4x4 matrices rather than 3x3. 3D Transformations Very similar to 2D. Using 4x4 matrices rather than 3x3. Translation

General 3D Rotation Translate the object such that rotation axis passes through the origin. Rotate the object such that rotation axis coincides with one of Cartesian axes. Perform specified rotation about the Cartesian axis. Apply inverse rotation to return rotation axis to original direction. Apply inverse translation to return rotation axis to original position.

Efficient 3D Rotations by Quaternions

3D Scaling Enlarging object also moves it from origin

Scaling with respect to a fixed point (not necessarily of object)