2D computer graphics: Difference between revisions

'''2D computer graphics''' is the [[Computercomputer-generated imagery|computer-based]] generation of [[digital image]]s—mostly from two-dimensional models (such as [[2D geometric model]]s, text, and digital images) and by techniques specific to them. It may refer to the branch of [[computer science]] that comprises such techniques or to the models themselves.

[[File:Blit dot.gif|thumb|[[Raster graphics|Raster graphic]] [[Sprite (computer graphics)|sprite]]s (left) and masks (right)]]

In [[Euclidean geometry]], a '''[[translation' (geometry)]]'' moves every point a constant distance in a specified direction. A translation can be described as a [[Euclidean group|rigid motion]]: other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant [[vector space|vector]] to every point, or as shifting the [[Originorigin (mathematics)|origin]] of the [[coordinate system]]. A '''[[shift operator|translation operator']]'' is an [[operator (mathematics)|operator]] <math>T_\mathbf{\delta}</math> such that <math>T_\mathbf{\delta} f(\mathbf{v}) = f(\mathbf{v}+\mathbf{\delta}).</math>

If ''T'' is a translation, then the [[image (mathematics)|image]] of a subset ''A'' under the [[function (mathematics)|function]] ''T'' is the '''translatetranslation''' of ''A'' by ''T''. The translatetranslation of ''A'' by ''T''_'''v''' is often written ''A'' + '''v'''.

=== Translation ===<!-- This section is linked from [[Affineaffine transformation]]. -->

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 (mathmathematics)|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).<ref>Richard Paul, 1981, [https://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators], MIT Press, Cambridge, MA</ref>

To translate an object by a [[Vectorvector (geometry)|vector]] '''v''', each homogeneous vector '''p''' (written in homogeneous coordinates) would need to be multiplied by this '''translation matrix''':

In [[linear algebra]], a '''[[rotation matrix']]'' is a [[matrix (mathematics)|matrix]] that is used to perform a [[rotation (mathematics)|rotation]] in [[Euclidean space]].

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 of rotation|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 [[Axis-angle representation|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.

====Non-standard orientation of the coordinate system====

[[File:Clockwise rotation SVG.svg|thumb|A rotation through angle ''θ'' with non-standard axes.]]

If a standard [[Orientation (mathematicsspace)|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.<ref>{{Citation|url=http://www.w3.org/TR/SVG/coords.html#InitialCoordinateSystem|title=Scalable Vector Graphics -- the initial coordinate system|authorwebsite=W3C recommendationw3.org|year=2003}}</ref>

See [[Rotation matrix#Ambiguities|below]] for other alternative conventions which may change the sense of the rotation produced by a [[rotation matrix]].

=====Common rotations=====

A scalingScaling 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 (linear algebra)|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.

====Using homogeneous coordinates====

Layered models are sometimes called "2{{frac|1|2}}-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 editor]]s. 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-oddeven–odd rule]].

Classic 2D [[graphics chip]]s and [[graphics processing unit]]s of the late 1970s to 1980s, used in [[8-bit computing|8-bit]] to early [[16-bit computing|16-bit]], [[arcade game]]s, [[video game console]]s, and [[home computer]]s, include:

*[[Atari, Inc.]]'s [[Television Interface Adapter|TIA]], [[ANTIC]], [[Color Television Interface Adapter|CTIA]] and [[George's Television Interface Adapter|GTIA]]

*[[Texas Instruments]]' [[Texas Instruments TMS9918|TMS9918]]

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 game]]s; and are still used for card and board games such as [[Solitaire (game)|solitaire]], [[chess]], [[mahjongg]], etc.

[[Digital image processing|Image editors]] are specialized for the manipulation of [[digital image]]s, 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 Computer|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<ref name="Pile">{{cite book |last=Pile Jr |first=John Jr. |author-link=John Pile Jr |date=May 2013 |title=2D Graphics Programming for Games |url=http://www.crcpress.com/product/isbn/9781466501898 |location=New York, NY |publisher=CRC Press |isbn=978-1466501898 }}</ref>{{rp|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|date=April 2013}} With software like [[RETAS]] [[UbiArt Framework]] and [[Adobe After Effects]], coloring and compositing can be done in less time.{{citation needed|date=April 2013}}

Programs like [[Blender (software)|Blender]] or [[Adobe Substance]] allow the user to do either 3D animation, 2D animation or combine both in its software allowing experimentation with multiple forms of animation.<ref>{{Cite web|url=https://www.blender.org/|title=blender.org - Home of the Blender project - Free and Open 3D Creation Software|last=Foundation|first=Blender|website=blender.org|language=en|access-date=2019-04-24}}</ref>