Displacement"'''''''''''''''''''''''''''''''''''''''''''''''''
Displacement may refer to:
Displacement (vector), in Newtonian mechanics, specifies the position of a point or a particle in reference to an origin or to a previous position
Displacement field (mechanics), as used in the continuum body mechanics
Electric displacement field, as appears in Maxwell's equations
Engine displacement, the total volume of air/fuel mixture an engine can draw in during one complete engine cycle
Displacement (fencing), a movement that avoids or dodges an attack
Displacement (fluid), an object immersed in a fluid pushes it out of the way. Ship measurement
Displacement hull, where the moving hull's weight is supported by buoyancy alone and it must displace water from its path rather than planing on the water's surface
Forced migration, by persecution or violence
Displacement (psychology), a sub-conscious defense mechanism
Displacement (psiology, parapsychology, psychical science), a statistical or qualitative correspondence between targets and responses
Single or double displacement reaction, a chemical reaction concerning the exchange of ions
Displacement (Orthopedic surgery), change in alignment of the fracture fragments
Earth Crustal Displacement, the shifting of earth's crust as a whole
Particle displacement is a measurement of distance of the movement of a particle in a medium (air) as it transmits a wave.
''Distance''''''''''''''''''''''''''''''''''''''''''''''''''''''''
Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, distance must meet more rigorous criteria.
Distances between objects and between non-empty sets
Various distance definitions are possible between objects. For example, between celestial bodies one should not confuse the surface-to-surface distance and the center-to-center distance. If the former is much less than the latter, as for a LEO, the first tends to be quoted (altitude), otherwise, e.g. for the Earth-Moon distance, the latter.
In general one can define the distance between two non-empty subsets of a given set as the infimum of the distances between any two of their respective points, which is the every-day meaning of the word. However, this does not define a metric, since with this definition the distance between two different but overlapping sets is zero. A definition that provides a metric defines the distance as the larger of two values, one being the supremum, for a point ranging over one set, of the infimum, for a second point ranging over the other set, of the distance between the points, and the other value being likewise defined but with the roles of the two sets swapped. This is called the Hausdorff metric.
[edit] Distance versus displacement
Distance along a path compared with displacementDistance cannot be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction.
The distance covered by a vehicle (often recorded by an odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point.
'''Scalar'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
[edit] Etymology
The word scalar derives from the English word "scale" for a range of numbers, which in turn is derived from scala (Latin for "ladder"). According to a citation in the Oxford English Dictionary the first recorded usage of the term was by W. R. Hamilton in 1846, to refer to the real part of a quaternion:
The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part.
[edit] Definitions and properties
[edit] Scalars of vector spaces
A vector space is defined as a set of vectors, a set of scalars, and a scalar multiplication operation that takes a scalar k and a vector v to another vector kv. For example, in a coordinate space, the scalar multiplication k(v1,v2,...,vn) yields (kv1,kv2,...,kvn). In a (linear) function space, kf is the function x k(f(x)).
The scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields.
[edit] Scalars as vector components
According to a fundamental theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a coordinate vector space where the coordinates are elements of K. For example, every real vector space of dimension n is isomorphic to n-dimensional real space Rn.
[edit] Scalar product
A scalar product space is a vector space V with an additional scalar product (or inner product) operation which allows two vectors to be multiplied to produce a number. The result is usually defined to be a member of V's scalar field. Since the inner product of a vector and itself has to be non-negative, a scalar product space can be defined only over fields that support the notion of sign. This excludes finite fields, for instance.
The existence of the scalar product makes it possible to carry geometric intuition over from Euclidean space by providing a well-defined notion of the angle between two vectors, and in particular a way of expressing when two vectors are orthogonal. Most scalar product spaces can also be considered normed vector spaces in a natural way.
[edit] Scalars in normed vector spaces
Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar ||v||. By definition, multiplying v by a scalar k also multiplies its norm by |k|. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k. A vector space equipped with a norm is called a normed vector space (or normed linear space).
The norm is usually defined to be an element of V's scalar field K, which restricts the latter to fields that support the notion of sign. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four arithmetic operations; thus the rational numbers Q are excluded, but the surd field is acceptable. For this reason, not every scalar product space is a normed vector space.
[edit] Scalars in modules
When the requirement that the set of scalars form a field is relaxed so that it need only form a ring (so that, for example, the division of scalars need not be defined), the resulting more general algebraic structure is called a module.
In this case the "scalars" may be complicated objects. For instance, if R is a ring, the vectors of the product space Rn can be made into a module with the n×n matrices with entries from R as the scalars. Another example comes from manifold theory, where the space of sections of the tangent bundle forms a module over the algebra of real functions on the manifold.
[edit] Scaling transformation
The scalar multiplication of vector spaces and modules is a special case of scaling, a kind of linear transformation.
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.
More generally, the scalars associated with a vector space may be complex numbers or elements from any algebraic field.
Also, a scalar product operation (not to be confused with scalar multiplication) may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called a inner product space.
The real component of a quaternion is also called its scalar part.
The term is also sometimes used informally to mean a vector, matrix, tensor, or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, which is formally a 1×1 matrix, is often said to be a scalar.
The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.
'Vector'''''"""""""""""""""''''''''''''''''''''''''''''''
Vector graphics (also called geometric modeling or object-oriented graphics) is the use of geometrical primitives such as points, lines, curves, and polygons, which are all based upon mathematical equations to represent images in computer graphics. It is used by contrast to the term raster graphics, which is the representation of images as a collection of pixels (dots).
Overview
All major modern current computer video displays translate vector representations of an image to a raster format. The drawing software is used for creating and editing vector graphics. You can change the image by editing these objects. Youcan stretch them, twist them, colour them and so on with a series of tools. The raster image, containing a value for every pixel on the screen, is stored in memory. Starting in the earliest days of computing in the 1950s and into the 1980s, a different type of display, the vector graphics system, was used. In these "calligraphic" systems the electron beam of the CRT display monitor was steered directly to trace out the shapes required, line segment by line segment, with the rest of the screen remaining black. This process was repeated many times a second ("stroke refresh") to achieve a flicker-free or near flicker-free picture. These systems allowed very high-resolution line art and moving images to be displayed without the (for that time) unthinkably huge amounts of memory that an equivalent-resolution raster system would have needed, and allowed entire subpictures to be moved, rotated, blinked, etc. by modifying only a few words of the graphic data "display file." These vector-based monitors were also known as X-Y displays.
A special type of vector display is known as the storage tube, which has a video tube that operates very similar to an Etch A Sketch. As the electron beam moves across the screen, an array of small low-power electron flood guns keep the path of the beam continuously illuminated. This allows the video display itself to act as a memory storage for the computer. The detail and resolution of the image can be very high, and the vector computer could slowly paint out paragraphs of text and complex images over a period of a few minutes, while the storage display kept the previously written parts continuously visible. The image retention of a storage display can last for many hours with the vector storage display powered, but the screen can clear instantly with the push of a button or a signal from the driving vector computer.
Vectorising is good for removing unnecessary detail from a photograph. This is especially useful for information graphics or line art. (Images were converted to JPEG for display on this page.)]]
An original photograph, a JPEG raster image.
Vectorising is good for reducing file sizes and for allowing for better scaling while retaining enough information for aesthetic appeal and, often, photorealism. Many vector graphic editors can automatically convert from raster to vector graphics, though this image was done manually.One of the first uses of vector graphic displays was the US SAGE air defense system. Vector graphics systems were only retired from U.S. en route air traffic control in 1999, and are likely still in use in military and specialised systems. Vector graphics were also used on the TX-2 at the MIT Lincoln Laboratory by computer graphics pioneer Ivan Sutherland to run his program Sketchpad in 1963.
Subsequent vector graphics systems include Digital's GT40 [1]. There was a home gaming system that used vector graphics called Vectrex as well as various arcade games like Asteroids and Space Wars. Storage scope displays, such as the Tektronix 4014, could also create dynamic vector images by driving the display at a lower intensity.
Modern vector graphics displays can be sometimes be found at laser light shows, using two fast-moving X-Y mirrors to rapidly draw shapes and text on a large screen.
The term vector graphics is mainly used today in the context of two-dimensional computer graphics. It is one of several modes an artist can use to create an image on a raster display. Other modes include text, multimedia and 3D rendering. Virtually all modern 3D rendering is done using extensions of 2D vector graphics techniques. Plotters used in technical drawing still draw vectors directly to paper.
[edit] Motivation
For example, consider circle of radius r. The main pieces of information a program needs in order to draw this circle are
that the following data are describing a circle
the radius r and equation of a circle
the location of the center point of the circle
stroke line style and colour (possibly transparent)
fill style and colour (possibly transparent)
Advantages to this style of drawing over raster graphics:
This minimal amount of information translates to a much smaller file size compared to large raster images (the size of representation doesn't depend on the dimensions of the object).
Correspondingly, one can indefinitely zoom in on e.g. a circle arc, and it remains smooth. On the other hand, a polygon representing a curve will reveal being not really curved.
On zooming in, lines and curves need not get wider proportionally. Often the width is either not increased or less than proportional. On the other hand, irregular curves represented by simple geometric shapes may be made proportionally wider when zooming in, to keep them looking smooth and not like these geometric shapes.
The parameters of objects are stored and can be later modified. This means that moving, scaling, rotating, filling etc. doesn't degrade the quality of a drawing. Moreover, it is usual to specify the dimensions in device-independent units, which results in the best possible rasterization on raster devices.
From a 3-D perspective, rendering shadows is also much more realistic with vector graphics, as shadows can be abstracted into the rays of light which form them. This allows for photo realistic images and renderings.