Information about Scale (measurement)
The concept of scale is applicable if a system is represented proportionally by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a dimensionless scale, e.g. 1:25; this scale is larger than 1:50.
In the general case of a differentiable bijection, the concept of scale can, to some extent, still be used, but it may depend on location and direction. It can be described by the Jacobian matrix. The modulus of the matrix times a unit vector is the scale in that direction. The non-linear case applies for example if a curved surface like part of the Earth's surface is mapped to a plane, see map projection.
In the case of an affine transformation the scale does not depend on location but it depends in general on direction. If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix, we have directionally differential scaling and the diagonal elements (the eigenvalues) are the scale factors in two or three perpendicular directions. For example, on some profile maps horizontal and vertical scale are different; in particular elevation may be shown in a larger scale than horizontal distance.
In the case of directional scaling (in one direction only) there is just one scale factor for one direction.
The case of uniform scaling corresponds to a geometric similarity. There is just one scale throughout.
In the case of an isometry the scale is 1:1.
In the more general case of one quantity represented by another one, the scale has also a physical dimension. E.g., if an arrow is drawn to represent a physical vector, the "scale" has a physical dimension equal to that of the vector, divided by length. For example, if a force of 1 newton is represented by an arrow of 2 cm, the scale is 1 m : 50 N. There is typically consistency in scale among quantities of the same dimension, but otherwise scales within the same diagram may vary; e.g "5 m" may also be represented by an arrow of 2 cm; in that case the scale for vectors which represent length is 1:250. Correspondingly, torques could be represented on the same map by areas in a scale of 1 m² : 12 500 Nm, which is equal to 1 m : 12 500 N. Torques in the plane of the map could be represented by arrows with an independent scale of e.g. 1 m : 300 Nm.
The scale of a map or enlarged or reduced model indicates the ratio between the distances on the map or model and the corresponding distances in reality or the original. E.g. a map of scale 1:50,000 shows a distance of 50,000 cm (=500 m) as 1 cm on a map, and a model on a scale 1:25 of a building with a height of 30 m has a model height of 1.20 m. An alternative method of indicating the scale is by a scale bar. This can also be applied on a computer screen etc., where the ratio may vary, and also remains valid when enlarging or reducing a paper map.
In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.
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In the general case of a differentiable bijection, the concept of scale can, to some extent, still be used, but it may depend on location and direction. It can be described by the Jacobian matrix. The modulus of the matrix times a unit vector is the scale in that direction. The non-linear case applies for example if a curved surface like part of the Earth's surface is mapped to a plane, see map projection.
In the case of an affine transformation the scale does not depend on location but it depends in general on direction. If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix, we have directionally differential scaling and the diagonal elements (the eigenvalues) are the scale factors in two or three perpendicular directions. For example, on some profile maps horizontal and vertical scale are different; in particular elevation may be shown in a larger scale than horizontal distance.
In the case of directional scaling (in one direction only) there is just one scale factor for one direction.
The case of uniform scaling corresponds to a geometric similarity. There is just one scale throughout.
In the case of an isometry the scale is 1:1.
In the more general case of one quantity represented by another one, the scale has also a physical dimension. E.g., if an arrow is drawn to represent a physical vector, the "scale" has a physical dimension equal to that of the vector, divided by length. For example, if a force of 1 newton is represented by an arrow of 2 cm, the scale is 1 m : 50 N. There is typically consistency in scale among quantities of the same dimension, but otherwise scales within the same diagram may vary; e.g "5 m" may also be represented by an arrow of 2 cm; in that case the scale for vectors which represent length is 1:250. Correspondingly, torques could be represented on the same map by areas in a scale of 1 m² : 12 500 Nm, which is equal to 1 m : 12 500 N. Torques in the plane of the map could be represented by arrows with an independent scale of e.g. 1 m : 300 Nm.
The scale of a map or enlarged or reduced model indicates the ratio between the distances on the map or model and the corresponding distances in reality or the original. E.g. a map of scale 1:50,000 shows a distance of 50,000 cm (=500 m) as 1 cm on a map, and a model on a scale 1:25 of a building with a height of 30 m has a model height of 1.20 m. An alternative method of indicating the scale is by a scale bar. This can also be applied on a computer screen etc., where the ratio may vary, and also remains valid when enlarging or reducing a paper map.
See also
- Scale (map)
- Scale (disambiguation)
- Scales of scale models
- Scale factor
- Scaling in gravity
- proportionality, see Proportionality (disambiguation).
In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.
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scale model is a representation or copy of an object that is larger or smaller than the actual size of the object being represented. Very often the scale model is smaller than the original and used as a guide to making the object in full size.
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In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number.
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In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
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f(x) = y.
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Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.
In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
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In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
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Views
Graphical projections
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Graphical projections
- Perspective projection
- Parallel projection
- Orthographic projection
- Plan, or floor plan view
- Section
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In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:
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In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero.
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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.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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A scale factor is a number which scales, or multiplies, some quantity. In the equation , is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to . For example, doubling distances corresponds to a scale factor of 2 for distance.
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A scale factor is a number which scales, or multiplies, some quantity. In the equation , is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to . For example, doubling distances corresponds to a scale factor of 2 for distance.
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Similarity is some degree of symmetry in either analogy and resemblance between two or more concepts or objects. The notion of similarity rests either on exact or approximate repetitions of patterns in the compared items.
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isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent.
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Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities.
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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torque (or often called a moment) can informally be thought of as "rotational force" or "angular force" which causes a change in rotational motion. This force is defined by linear force multiplied by a radius.
The SI unit for torque is the newton meter (N m). In U.S.
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The SI unit for torque is the newton meter (N m). In U.S.
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The scale of a map is the ratio of a single unit of distance on the map to the equivalent distance on the ground. The scale can be expressed in four ways: as a ratio, a fraction, in words and as a graphical (bar) scale.
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model is used in various contexts to mean a physical representation of some thing. That thing may be a single item or object (for example, a bolt) or a large system (for example, the Solar System).
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1 centimetre =
SI units
010−3 m 0 mm
US customary / Imperial units
010−3 ft 0 in
A centimetre (American spelling: centimeter, symbol cmSI units
010−3 m 0 mm
US customary / Imperial units
010−3 ft 0 in
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1 metre =
SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in
The metre or meter[1](symbol: m) is the fundamental unit of length in the International System of Units (SI).SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in
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The scale of a map is the ratio of a single unit of distance on the map to the equivalent distance on the ground. The scale can be expressed in four ways: as a ratio, a fraction, in words and as a graphical (bar) scale.
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Scale can refer to:
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- Scale (computing). See also Scalability.
- Scale (map)
- Scale (ratio)
- Scale factor
- Scale (spatial)
- Scale (zoology)
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A scale factor is a number which scales, or multiplies, some quantity. In the equation , is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to . For example, doubling distances corresponds to a scale factor of 2 for distance.
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