2D computer graphics

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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. It may refer to the branch of computer science that comprises such techniques or to the models themselves.

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).

In many domains, such as

.

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.

2D graphics models may combine

. In .

Background (geometry)

This section splitting the content into a new article. (May 2022)

In Euclidean geometry, a translation (geometry) 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 ${\displaystyle T_{\mathbf {\delta } }}$ such that ${\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).}$

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 translation of A by T. The translation 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 ).

Translation

Since a translation is an

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:

${\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}}$

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

${\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} }$

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

${\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!}$

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

${\displaystyle T_{\mathbf {u} }T_{\mathbf {v} }=T_{\mathbf {u} +\mathbf {v} }.\!}$

Because addition of vectors is

commutative

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

Rotation

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

${\displaystyle R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}$

rotates points in the xy-

Rv. 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

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:

${\displaystyle R^{T}=R^{-1},\det R=1\,}$.

The

SO(n).

In two dimensions

In two dimensions every rotation matrix has the following form:

${\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}$.

This rotates

:

${\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}}$.

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

${\displaystyle x'=x\cos \theta -y\sin \theta \,}$,

${\displaystyle y'=x\sin \theta +y\cos \theta \,}$.

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

${\displaystyle R(-\theta )={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}\,}$.

Non-standard orientation of the coordinate system

If a standard

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

Common rotations

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

${\displaystyle R(90^{\circ })={\begin{bmatrix}0&-1\\[3pt]1&0\\\end{bmatrix}}}$ (90° counterclockwise rotation)

${\displaystyle R(180^{\circ })={\begin{bmatrix}-1&0\\[3pt]0&-1\\\end{bmatrix}}}$ (180° rotation in either direction – a half-turn)

${\displaystyle R(270^{\circ })={\begin{bmatrix}0&1\\[3pt]-1&0\\\end{bmatrix}}}$ (270° counterclockwise rotation, the same as a 90° clockwise rotation)

Scaling

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In

(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 (

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

scaling matrix

:

${\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}$

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

${\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}$

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 enlargement or dilation by a factor k, increasing the area by a factor of k² and the volume by a factor of k³.

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.

Using homogeneous coordinates

In

matrix:

${\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}$

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

${\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}$

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:

${\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}$

For each vector p = (p_x, p_y, p_z, 1) we would have

${\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}}}$

which would be homogenized to

${\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}$

Techniques

Direct painting

A convenient way to create a complex image is to start with a blank "canvas"

.

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
  arc
  , or simple curve with a virtual pen of given width.

Extended color models

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

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.

Layers

The models used in 2D computer graphics usually do not provide for three-dimensional shapes, or three-dimensional optical phenomena such as lighting,

—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 "21⁄2-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

.

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

artifacts wherever two elements overlap the same pixel.

See also

Portable Document Format#Layers

.

Hardware

Modern computer

chip.

Classic 2D

, include:

• TIA, ANTIC, CTIA and GTIA
• Capcom's CPS-A and CPS-B
• OCS
• MOS Technology's VIC and VIC-II
• Hudson Soft's Cynthia and HuC6270
• NEC's μPD7220 and μPD72120
• S-PPU
• 315-5196/315-5197
• Texas Instruments' TMS9918
• YM7101 VDP

Software

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

, 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,

. 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.

, etc.

Developmental animation

With the resurgence[4]^: 8 of 2D animation, free and proprietary software packages have become widely available for amateurs and professional animators. 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

.

Programs like 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.^[5]

See also

• 2.5D
• 3D computer graphics
• Computer animation
• CGI
• Bit blit
• Computer graphics
• Graphic art software
• Graphics
• Image scaling
• List of home computers by video hardware
• Turtle graphics
• Transparency in graphics
• Palette (computing)
• Pixel art

References