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In Euclidean geometry, **uniform scaling** (or ** isotropic scaling**^{ [1] }) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. 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.

- Matrix representation
- Scaling in arbitrary dimensions
- Using homogeneous coordinates
- Function dilation and contraction
- Particular cases
- See also
- Footnotes
- External links

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.

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**.

In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which 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).

Scaling is a linear transformation, and a special case of homothetic transformation. In most cases, the homothetic transformations are non-linear transformations.

A scaling can be represented by a scaling matrix. To scale an object by a vector *v* = (*v _{x}, v_{y}, v_{z}*), each point

As shown below, the multiplication will give the expected result:

Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three.

The scaling is uniform if and only if the scaling factors are equal (*v _{x} = v_{y} = v_{z}*). If all except one of the scale factors are equal to 1, we have directional scaling.

In the case where *v _{x} = v_{y} = v_{z} = k*, scaling increases the area of any surface by a factor of

In -dimensional space , uniform scaling by a factor is accomplished by scalar multiplication with , that is, multiplying each coordinate of each point by . As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a diagonal matrix whose entries on the diagonal are all equal to , namely .

Non-uniform scaling is accomplished by multiplication with any symmetric matrix. The eigenvalues of the matrix are the scale factors, and the corresponding eigenvectors are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis by the factor .

In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue.

In projective geometry, often used in computer graphics, points are represented using homogeneous coordinates. To scale an object by a vector *v* = (*v _{x}, v_{y}, v_{z}*), each homogeneous coordinate vector

As shown below, the multiplication will give the expected result:

Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor *s* (uniform scaling) can be accomplished by using this scaling matrix:

For each vector *p* = (*p _{x}, p_{y}, p_{z}*, 1) we would have

which would be equivalent to

Given a point , the dilation associates it with the point through the equations for .

Therefore, given a function , the equation of the dilated function is

If , the transformation is horizontal; when , it is a dilation, when , it is a contraction.

If , the transformation is vertical; when it is a dilation, when , it is a contraction.

If or , the transformation is a squeeze mapping.

- ↑ Durand; Cutler. "Transformations" (PowerPoint). Massachusetts Institute of Technology. Retrieved 12 September 2008.

Wikimedia Commons has media related to . Scaling (geometry) |

- Understanding 2D Scaling and Understanding 3D Scaling by Roger Germundsson, The Wolfram Demonstrations Project.

In physics, the **Lorentz transformations** are a one-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parametrized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

**2D computer graphics** is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them. The word may stand for the branch of computer science that comprises such techniques or for the models themselves.

In Euclidean geometry, an **affine transformation**, or an **affinity**, is a geometric transformation that preserves lines and parallelism.

In linear algebra, the **column space** of a matrix *A* is the span of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.

In linear algebra, an **orthogonal matrix** is a square matrix whose columns and rows are orthogonal unit vectors.

In linear algebra, the **singular value decomposition** (**SVD**) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition.

In vector calculus, the **Jacobian matrix** of a vector-valued function in several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the **Jacobian determinant**. Both the matrix and the determinant are often referred to simply as the **Jacobian** in literature.

**Orthographic projection** is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are *not* orthogonal to the projection plane.

In Euclidean geometry, a **translation** is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

In physics and mathematics, the **Lorentz group** is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In mathematics, a **homothety** is a transformation of an affine space determined by a point *S* called its *center* and a nonzero number *λ* called its *ratio*, which sends

In linear algebra and functional analysis, a **projection** is a linear transformation from a vector space to itself such that . That is, whenever is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then

In linear algebra, a **rotation matrix** is a matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

In computer graphics, **back-face culling** determines whether a polygon of a graphical object is visible. It is a step in the graphical pipeline that tests whether the points in the polygon appear in clockwise or counter-clockwise order when projected onto the screen. If the user has specified that front-facing polygons have a clockwise winding, but the polygon projected on the screen has a counter-clockwise winding then it has been rotated to face away from the camera and will not be drawn.

In linear algebra, an **eigenvector** or **characteristic vector** of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding **eigenvalue** is the factor by which the eigenvector is scaled.

In geometry, a **point reflection** or **inversion in a point** is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess **point symmetry**; if it is invariant under point reflection through its center, it is said to possess **central symmetry** or to be **centrally symmetric.**

In linear algebra, **eigendecomposition** or sometimes **spectral decomposition** is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way.

The **direct-quadrature-zero****transformation** or **zero-direct-quadrature****transformation** is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.

There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.

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