Difference between revisions of "Scaling (geometry)"
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Revision as of 08:20, 28 February 2016
In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.
Contents
Uniform scaling
The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar.
Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
Non-uniform scaling
More general is scaling with a separate scale factor for each axis direction.
Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction).
Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles).
It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it.
Enlargement and contraction
When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement.
When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction.
Special cases
In the most general sense, a scaling includes the case that the directions of scaling are not perpendicular.
It includes also the case that one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection).
Momothetic transformations
Scaling is a linear transformation, and a special case of homothetic transformation.
In most cases, the homothetic transformations are non-linear transformations.
See also
External links
- Scaling (geometry) @ Wikipedia
