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Encyclopedia > Scaling (geometry)
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In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. The result of uniform scaling is similar (in the geometric sense) to the original. Jump to: navigation, search In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Jump to: navigation, search A scale factor is a number which scales some quantity. ... In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ... Several equivalence relations in mathematics are called similarity. ...


More general is scaling with a separate scale factor for each axis direction; a special case is directional scaling (in one direction). Shapes may change; e.g. a rectangle may change into a rectangle of a different shape, but also in a parallelogram (the angles between lines parallel to the axes are preserved, but not all angles). In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ...


A scaling can be represented by a scaling matrix. 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: In mathematics, and in particular in vectorial analysis a vector is an arrow pointing from one point to another. ...

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. For the geometric term, see diameter. ... This article explains the meaning of area as a physical quantity. ... Jump to: navigation, search Volume, also called capacity, is a quantification of how much space an object occupies. ...


A scaling in the most general sense is any affine transformation with a diagonalizable matrix. It includes the case that the three 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. The latter corresponds to a combination of scaling proper and a kind of reflection: along lines in a particular direction we take the reflection in the point of intersection with a plane that need not be perpendicular; therefore it is more general than ordinary reflection in the plane. In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ...


Often, it is more useful to use homogeneous coordinates, since translation cannot be accomplished with a 3-by-3 matrix. To scale an object by a vector v = (vx, vy, vz), each homogeneous vector p = (px, py, pz, 1) would need to be multiplied with this scaling matrix: In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In mathematics, and in particular in vectorial analysis a vector is an arrow pointing from one point to another. ... In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ...

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

The scaling is uniform iff the scaling factors are equal. If all scale factors except one are 1 we have directional scaling. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...


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

For each homogeneous vector p = (px, py, pz, 1) we would have In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ...

which would be homogenized to


  Results from FactBites:
 
scaling - Search Results - MSN Encarta (366 words)
Scale (music) (Italian scala, “ladder”), the arrangement, by rising or falling pitch sequence, of the tones used in a musical system.
Scaling may refer to: * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scaling (computer network), a network's ability to function as the number of people on the...
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety.
Isner (2450 words)
With scaling the behavior is a bit different than a simple delta that starts at zero and becomes your values at the end of the stick.
You are scaling back to the inverse of the scaling only, and that results in a non-uniformly scaled (but not sheared) starting grid on the inverse.
Basically if you have scaling on the origin grid (which is what introduces shearing) then we need to add two add two new steps to the visual instructions to also invert that origin scale.
  More results at FactBites »

 
 

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