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**2D computer graphics** is the computer-based generation of digital images —mostly from two-dimensional models (such as 2D geometric models, text, and digital images) 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.

A **computer** is a machine that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of operations, called *programs.* These programs enable computers to perform an extremely wide range of tasks. A "complete" computer including the hardware, the operating system, and peripheral equipment required and used for "full" operation can be referred to as a **computer system**. This term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster.

A **digital image** is a numeric representation, normally binary, of a two-dimensional image. Depending on whether the image resolution is fixed, it may be of vector or raster type. By itself, the term "digital image" usually refers to raster images or bitmapped images.

A **2D geometric model** is a geometric model of an object as a two-dimensional figure, usually on the Euclidean or Cartesian plane.

- 2D graphics techniques
- Translation
- Rotation
- In two dimensions
- Non-standard orientation of the coordinate system
- Common rotations
- Scaling
- Using homogeneous coordinates
- Direct painting
- Extended color models
- Layers
- 2D graphics hardware
- 2D graphics software
- Developmental animation
- See also
- References

2D computer graphics are mainly used in applications that were originally developed upon traditional printing and drawing technologies, such as typography, cartography, technical drawing, advertising, etc. In those applications, the two-dimensional image is not just a representation of a real-world object, but an independent artifact with added semantic value; two-dimensional models are therefore preferred, because they give more direct control of the image than 3D computer graphics (whose approach is more akin to photography than to typography).

**Printing** is a process for reproducing text and images using a master form or template. The earliest non-paper products involving printing include cylinder seals and objects such as the Cyrus Cylinder and the Cylinders of Nabonidus. The earliest known form of printing as applied to paper was woodblock printing, which appeared in China before 220 AD. Later developments in printing technology include the movable type invented by Bi Sheng around 1040 AD and the printing press invented by Johannes Gutenberg in the 15th century. The technology of printing played a key role in the development of the Renaissance and the scientific revolution, and laid the material basis for the modern knowledge-based economy and the spread of learning to the masses.

**Drawing** is a form of visual art in which a person uses various drawing instruments to mark paper or another two-dimensional medium. Instruments include graphite pencils, pen and ink, various kinds of paints, inked brushes, colored pencils, crayons, charcoal, chalk, pastels, various kinds of erasers, markers, styluses, and various metals. Digital drawing is the act of using a computer to draw. Common methods of digital drawing include a stylus or finger on a touchscreen device, stylus- or finger-to-touchpad, or in some cases, a mouse. There are many digital art programs and devices.

**Typography** is the art and technique of arranging type to make written language legible, readable, and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing (leading), and letter-spacing (tracking), and adjusting the space between pairs of letters (kerning). The term *typography* is also applied to the style, arrangement, and appearance of the letters, numbers, and symbols created by the process. Type design is a closely related craft, sometimes considered part of typography; most typographers do not design typefaces, and some type designers do not consider themselves typographers. Typography also may be used as a decorative device, unrelated to communication of information.

In many domains, such as desktop publishing, engineering, and business, a description of a document based on 2D computer graphics techniques can be much smaller than the corresponding digital image—often by a factor of 1/1000 or more. This representation is also more flexible since it can be rendered at different resolutions to suit different output devices. For these reasons, documents and illustrations are often stored or transmitted as 2D graphic files.

**Desktop publishing** (**DTP**) is the creation of documents using page layout software on a personal ("desktop") computer. It was first used almost exclusively for print publications, but now it also assists in the creation of various forms of online content. Desktop publishing software can generate layouts and produce typographic-quality text and images comparable to traditional typography and printing. Desktop publishing is also the main reference for **digital typography**. This technology allows individuals, businesses, and other organizations to self-publish a wide variety of content, from menus to magazines to books, without the expense of commercial printing.

**Engineering** is the use of scientific principles to design and build machines, structures, and other things, including bridges, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering.

**Business** is the activity of making one's living or making money by producing or buying and selling products. Simply put, it is "any activity or enterprise entered into for profit. It does not mean it is a company, a corporation, partnership, or have any such formal organization, but it can range from a street peddler to General Motors."

2D computer graphics started in the 1950s, based on vector graphics devices. These were largely supplanted by raster-based devices in the following decades. The PostScript language and the X Window System protocol were landmark developments in the field.

**Vector graphics** are computer graphics images that are defined in terms of 2D points, which are connected by lines and curves to form polygons and other shapes. Each of these points has a definite position on the *x-* and *y-*axis of the work plane and determines the direction of the path; further, each path may have various properties including values for stroke color, shape, curve, thickness, and fill. Vector graphics are commonly found today in the SVG, EPS, PDF or AI graphic file formats and are intrinsically different from the more common raster graphics file formats of JPEG, PNG, APNG, GIF, and MPEG4.

In computer graphics, a **raster graphics** or **bitmap** image is a dot matrix data structure that represents a generally rectangular grid of pixels, viewable via a monitor, paper, or other display medium. Raster images are stored in image files with varying formats.

**PostScript** (**PS**) is a page description language in the electronic publishing and desktop publishing business. It is a dynamically typed, concatenative programming language and was created at Adobe Systems by John Warnock, Charles Geschke, Doug Brotz, Ed Taft and Bill Paxton from 1982 to 1984.

2D graphics models may combine geometric models (also called vector graphics), digital images (also called raster graphics), text to be typeset (defined by content, font style and size, color, position, and orientation), mathematical functions and equations, and more. These components can be modified and manipulated by two-dimensional geometric transformations such as translation, rotation, scaling. In object-oriented graphics, the image is described indirectly by an object endowed with a self-rendering method —a procedure which assigns colors to the image pixels by an arbitrary algorithm. Complex models can be built by combining simpler objects, in the paradigms of object-oriented programming.

**Typesetting** is the composition of text by means of arranging physical types or the digital equivalents. Stored letters and other symbols are retrieved and ordered according to a language's orthography for visual display. Typesetting requires one or more fonts. One significant effect of typesetting was that authorship of works could be spotted more easily, making it difficult for copiers who have not gained permission.

In typography, a **typeface** is a set of one or more fonts each composed of glyphs that share common design features. Each font of a typeface has a specific weight, style, condensation, width, slant, italicization, ornamentation, and designer or foundry. For example, "ITC Garamond Bold Condensed Italic" means the bold, condensed-width, italic version of ITC Garamond. It is a different font from "ITC Garamond Condensed Italic" and "ITC Garamond Bold Condensed", but all are fonts within the same typeface, "ITC Garamond". ITC Garamond is a different typeface from "Adobe Garamond" or "Monotype Garamond". There are thousands of different typefaces in existence, with new ones being developed constantly.

In mathematics, a **function** is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.

In Euclidean geometry, a **translation** moves every point a constant distance in a specified direction. A translation can be described as a rigid motion: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A **translation operator** is an operator such that

**Euclidean geometry** is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The *Elements* begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the *Elements* states results of what are now called algebra and number theory, explained in geometrical language.

In mathematics, an **Euclidean group** is the group of (Euclidean) isometries of an Euclidean space 𝔼^{n}; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension *n* of the space, and is commonly denoted E(*n*) or ISO(*n*).

A **vector space** is a collection of objects called **vectors**, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called *axioms*, listed below.

If **v** is a fixed vector, then the translation *T*_{v} will work as *T*_{v}(**p**) = **p** + **v**.

If *T* is a translation, then the image of a subset *A* under the function *T* is the **translate** of *A* by *T*. The translate of *A* by *T*_{v} is often written *A* + **v**.

In a Euclidean space, any translation is an isometry. The set of all translations forms the translation group *T*, which is isomorphic to the space itself, and a normal subgroup of Euclidean group *E*(*n* ). The quotient group of *E*(*n* ) by *T* is isomorphic to the orthogonal group *O*(*n* ):

*E*(*n*)*/ T*≅*O*(*n*).

Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear. Thus we write the 3-dimensional vector **w** = (*w*_{x}, *w*_{y}, *w*_{z}) using 4 homogeneous coordinates as **w** = (*w*_{x}, *w*_{y}, *w*_{z}, 1).^{ [1] }

To translate an object by a vector **v**, each homogeneous vector **p** (written in homogeneous coordinates) would need to be multiplied by this **translation matrix**:

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

The inverse of a translation matrix can be obtained by reversing the direction of the vector:

Similarly, the product of translation matrices is given by adding the vectors:

Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

In linear algebra, a **rotation matrix** is a matrix that is used to perform a rotation in Euclidean space.

rotates points in the *xy*-Cartesian plane counterclockwise through an angle *θ* about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix *R*, the position of each point must be represented by a column vector **v**, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication *R***v**. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system.

Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In 2-dimensional space, a rotation can be simply described by an angle *θ* of rotation, but it can be also represented by the 4 entries of a rotation matrix with 2 rows and 2 columns. In 3-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with 3 entries. However, it can also be represented by the 9 entries of a rotation matrix with 3 rows and 3 columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a **rotational displacement**, which can be represented by a matrix, but no associated single axis or angle.

Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1:

- .

The set of all such matrices of size *n* forms a group, known as the special orthogonal group SO(*n*).

In two dimensions every rotation matrix has the following form:

- .

This rotates column vectors by means of the following matrix multiplication:

- .

So the coordinates (x',y') of the point (x,y) after rotation are:

- ,
- .

The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°).

- .

If a standard right-handed Cartesian coordinate system is used, with the *x* axis to the right and the *y* axis up, the rotation R(*θ*) is counterclockwise. If a left-handed Cartesian coordinate system is used, with *x* directed to the right but *y* directed down, R(*θ*) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the *y*-axis down the screen or page.^{ [2] }

See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.

Particularly useful are the matrices for 90° and 180° rotations:

- (90° counterclockwise rotation)
- (180° rotation in either direction – a half-turn)
- (270° counterclockwise rotation, the same as a 90° clockwise rotation)

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In Euclidean geometry, **uniform scaling** (** isotropic scaling**,^{ [3] }**homogeneous dilation**, homothety) 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. (Some school text books specifically exclude this possibility, just as some exclude squares from being rectangles or circles from being ellipses.)

More general is **scaling** with a separate scale factor for each axis direction. **Non-uniform scaling** (** anisotropic scaling**, **inhomogeneous dilation**) 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).

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*, the scaling is also called an

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 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 homogenized to

A convenient way to create a complex image is to start with a blank "canvas" raster map (an array of pixels, also known as a bitmap) filled with some uniform background color and then "draw", "paint" or "paste" simple patches of color onto it, in an appropriate order. In particular the canvas may be the frame buffer for a computer display.

Some programs will set the pixel colors directly, but most will rely on some 2D graphics library or the machine's graphics card, which usually implement the following operations:

- paste a given image at a specified offset onto the canvas;
- write a string of characters with a specified font, at a given position and angle;
- paint a simple geometric shape, such as a triangle defined by three corners, or a circle with given center and radius;
- draw a line segment, arc, or simple curve with a
*virtual pen*of given width.

Text, shapes and lines are rendered with a client-specified color. Many libraries and cards provide color gradients, which are handy for the generation of smoothly-varying backgrounds, shadow effects, etc. (See also Gouraud shading). The pixel colors can also be taken from a texture, e.g. a digital image (thus emulating rub-on screentones and the fabled *checker paint* which used to be available only in cartoons).

Painting a pixel with a given color usually replaces its previous color. However, many systems support painting with transparent and translucent colors, which only modify the previous pixel values. The two colors may also be combined in more complex ways, e.g. by computing their bitwise exclusive or. This technique is known as *inverting color* or *color inversion*, and is often used in graphical user interfaces for highlighting, rubber-band drawing, and other volatile painting—since re-painting the same shapes with the same color will restore the original pixel values.

The models used in 2D computer graphics usually do not provide for three-dimensional shapes, or three-dimensional optical phenomena such as lighting, shadows, reflection, refraction, etc. However, they usually can model multiple *layers* (conceptually of ink, paper, or film; opaque, translucent, or transparent —stacked in a specific order. The ordering is usually defined by a single number (the layer's *depth*, or distance from the viewer).

Layered models are sometimes called *2½-D computer graphics*. They make it possible to mimic traditional drafting and printing techniques based on film and paper, such as cutting and pasting; and allow the user to edit any layer without affecting the others. For these reasons, they are used in most graphics editors. Layered models also allow better spatial anti-aliasing of complex drawings and provide a sound model for certain techniques such as *mitered joints* and the even-odd rule.

Layered models are also used to allow the user to suppress unwanted information when viewing or printing a document, e.g. roads or railways from a map, certain process layers from an integrated circuit diagram, or hand annotations from a business letter.

In a layer-based model, the target image is produced by "painting" or "pasting" each layer, in order of decreasing depth, on the virtual canvas. Conceptually, each layer is first rendered on its own, yielding a digital image with the desired resolution which is then painted over the canvas, pixel by pixel. Fully transparent parts of a layer need not be rendered, of course. The rendering and painting may be done in parallel, i.e., each layer pixel may be painted on the canvas as soon as it is produced by the rendering procedure.

Layers that consist of complex geometric objects (such as text or polylines) may be broken down into simpler elements (characters or line segments, respectively), which are then painted as separate layers, in some order. However, this solution may create undesirable aliasing artifacts wherever two elements overlap the same pixel.

See also Portable Document Format#Layers.

Modern computer graphics card displays almost overwhelmingly use raster techniques, dividing the screen into a rectangular grid of pixels, due to the relatively low cost of raster-based video hardware as compared with vector graphic hardware. Most graphic hardware has internal support for blitting operations or sprite drawing. A co-processor dedicated to blitting is known as a * Blitter chip*.

Classic 2D graphics chips and graphics processing units of the late 1970s to 1980s, used in 8-bit to early 16-bit, arcade games, video game consoles, and home computers, include:

- Atari's TIA, ANTIC, CTIA and GTIA
- Capcom's CPS-A and CPS-B
- Commodore's OCS
- MOS Technology's VIC and VIC-II
- Fujitsu's MB14241
- Hudson Soft's Cynthia and HuC6270
- NEC's µPD7220 and µPD72120
- Ricoh's PPU and S-PPU
- Sega's VDP, Super Scaler, 315-5011/315-5012 and 315-5196/315-5197
- Texas Instruments' TMS9918
- Yamaha's V9938, V9958 and YM7101 VDP

Many graphical user interfaces (GUIs), including macOS, Microsoft Windows, or the X Window System, are primarily based on 2D graphical concepts. Such software provides a visual environment for interacting with the computer, and commonly includes some form of window manager to aid the user in conceptually distinguishing between different applications. The user interface within individual software applications is typically 2D in nature as well, due in part to the fact that most common input devices, such as the mouse, are constrained to two dimensions of movement.

2D graphics are very important in the control peripherals such as printers, plotters, sheet cutting machines, etc. They were also used in most early video games; and are still used for card and board games such as solitaire, chess, mahjongg, etc.

2D graphics editors or *drawing programs* are application-level software for the creation of images, diagrams and illustrations by direct manipulation (through the mouse, graphics tablet, or similar device) of 2D computer graphics primitives. These editors generally provide geometric primitives as well as digital images; and some even support procedural models. The illustration is usually represented internally as a layered model, often with a hierarchical structure to make editing more convenient. These editors generally output graphics files where the layers and primitives are separately preserved in their original form. MacDraw, introduced in 1984 with the Macintosh line of computers, was an early example of this class; recent examples are the commercial products Adobe Illustrator and CorelDRAW, and the free editors such as xfig or Inkscape. There are also many 2D graphics editors specialized for certain types of drawings such as electrical, electronic and VLSI diagrams, topographic maps, computer fonts, etc.

Image editors are specialized for the manipulation of digital images, mainly by means of free-hand drawing/painting and signal processing operations. They typically use a direct-painting paradigm, where the user controls virtual pens, brushes, and other free-hand artistic instruments to apply paint to a virtual canvas. Some image editors support a multiple-layer model; however, in order to support signal-processing operations like blurring each layer is normally represented as a digital image. Therefore, any geometric primitives that are provided by the editor are immediately converted to pixels and painted onto the canvas. The name *raster graphics editor* is sometimes used to contrast this approach to that of general editors which also handle *vector graphics*. One of the first popular image editors was Apple's MacPaint, companion to MacDraw. Modern examples are the free GIMP editor, and the commercial products Photoshop and Paint Shop Pro. This class too includes many specialized editors — for medicine, remote sensing, digital photography, etc.

With the resurgence^{ [4] }^{:8} of 2D animation, free and proprietary software packages have become widely available for amateurs and professional animators. The principal issue with 2D animation is labor requirements.^{[ citation needed ]} With software like RETAS UbiArt Framework and Adobe After Effects, coloring and compositing can be done in less time.^{[ citation needed ]}

Various approaches have been developed^{ [4] }^{:38} to aid and speed up the process of digital 2D animation. For example, by generating vector artwork in a tool like Adobe Flash an artist may employ software-driven automatic coloring and in-betweening.

Programs like blender allow the user to do either 3d animation, 2d animation or combine both in its software allowing you to experiment with multiple forms of animation seamlessly.^{ [5] }

Wikimedia Commons has media related to . 2D |

**Kinematics** is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

**Angular displacement** of a body is the angle in radians through which a point revolves around a centre or line has been rotated in a specified sense about a specified axis. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (*t*). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, i.e.

In mathematics, a **unit vector** in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as **d**. Two 2D direction vectors, **d1** and **d2** are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space **R**^{3} under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more compact, more numerically stable, and more efficient. Quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites and crystallographic texture analysis.

**3D projection** is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.

In mathematics, a function defined on an inner product space is said to have **rotational invariance** if its value does not change when arbitrary rotations are applied to its argument.

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

Note: This page uses common physics notation for spherical coordinates, in which is the angle between the *z* axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the *x-y* plane and the *x* axis. Several other definitions are in use, and so care must be taken in comparing different sources.

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 mathematics, the **Cayley transform**, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators.

In the theory of three-dimensional rotation, **Rodrigues' rotation formula**, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from **so**(3), the Lie algebra of SO(3), to SO(3) without actually computing the full matrix exponential.

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

This article derives the main properties of rotations in 3-dimensional space.

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.

In materials science, a **composite laminate** is an assembly of layers of fibrous composite materials which can be joined to provide required engineering properties, including in-plane stiffness, bending stiffness, strength, and coefficient of thermal expansion.

- ↑ Richard Paul, 1981, Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators, MIT Press, Cambridge, MA
- ↑ W3C recommendation (2003),
*Scalable Vector Graphics -- the initial coordinate system* - ↑ Durand; Cutler. "Transformations" (PowerPoint). Massachusetts Institute of Technology. Retrieved 12 September 2008.
- 1 2 Pile Jr, John (May 2013).
*2D Graphics Programming for Games*. New York, NY: CRC Press. ISBN 1466501898. - ↑ Foundation, Blender. "blender.org - Home of the Blender project - Free and Open 3D Creation Software".
*blender.org*. Retrieved 2019-04-24.

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