A geometrical transformation refers
to any operation that changes the position, size, shape, or orientation of a
geometric figure. These transformations can be classified into
different types based on how they alter the figure:
1. Translation: This shifts a figure from one location to another
without changing its shape, size, or orientation. Every point of the figure
moves the same distance in the same direction.
2. Rotation: This turns a figure around a fixed point called the
center of rotation. The shape and size of the figure remain unchanged, but its
orientation changes.
3. Reflection: This flips a figure over a line (called the line of
reflection), creating a mirror image. The shape and size of the figure remain
the same, but the figure's orientation is reversed.
4. Scaling (Dilation): This changes the size of a figure by enlarging or
shrinking it. The shape of the figure remains similar, but its size changes by
a scale factor.
5. Shearing: This shifts one part of a figure in a direction
parallel to a line while keeping the rest of the figure fixed or moving it in a
parallel manner. This transformation distorts the shape of the figure.
6. Affine Transformation: This is a combination of linear transformations (like
scaling, rotating, shearing) and translations. It preserves lines and
parallelism (i.e., parallel lines remain parallel).
7. Projective Transformation: This is a more complex transformation that can map
lines to lines but does not necessarily preserve parallelism or ratios of
lengths.
These transformations can be described using matrices and
vector operations in linear algebra, providing a powerful and concise way to
perform and analyze them in various applications, including computer graphics,
robotics, and image processing.
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