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.

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
T
δ
such that
T
δ
f
(
v
)
=
f
(
v
+
δ
)
.

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).

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**:

T
v
=
[
1
0
0
v
x
0
1
0
v
y
0
0
1
v
z
0
0
0
1
]
As shown below, the multiplication will give the expected result:

T
v
p
=
[
1
0
0
v
x
0
1
0
v
y
0
0
1
v
z
0
0
0
1
]
[
p
x
p
y
p
z
1
]
=
[
p
x
+
v
x
p
y
+
v
y
p
z
+
v
z
1
]
=
p
+
v
The inverse of a translation matrix can be obtained by reversing the direction of the vector:

T
v
−
1
=
T
−
v
.

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

T
u
T
v
=
T
u
+
v
.
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.

R
=
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
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:

R
T
=
R
−
1
,
det
R
=
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:

R
(
θ
)
=
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
.

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

[
x
′
y
′
]
=
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
[
x
y
]
.

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

x
′
=
x
cos
θ
−
y
sin
θ
,

y
′
=
x
sin
θ
+
y
cos
θ
.

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

R
(
−
θ
)
=
[
cos
θ
sin
θ
−
sin
θ
cos
θ
]
.

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.

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:

R
(
90
∘
)
=
[
0
−
1
1
0
]
(90° counterclockwise rotation)

R
(
180
∘
)
=
[
−
1
0
0
−
1
]
(180° rotation in either direction – a half-turn)

R
(
270
∘
)
=
[
0
1
−
1
0
]
(270° counterclockwise rotation, the same as a 90° clockwise rotation)

In Euclidean geometry, **uniform scaling** (**isotropic scaling**, **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 *p* = (*p*_{x}, p_{y}, p_{z}) would need to be multiplied with this scaling matrix:

S
v
=
[
v
x
0
0
0
v
y
0
0
0
v
z
]
.
As shown below, the multiplication will give the expected result:

S
v
p
=
[
v
x
0
0
0
v
y
0
0
0
v
z
]
[
p
x
p
y
p
z
]
=
[
v
x
p
x
v
y
p
y
v
z
p
z
]
.
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 k^{2} and the volume by a factor of k^{3}.

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 *p* = (*p*_{x}, p_{y}, p_{z}, 1) would need to be multiplied with this projective transformation matrix:

S
v
=
[
v
x
0
0
0
0
v
y
0
0
0
0
v
z
0
0
0
0
1
]
.
As shown below, the multiplication will give the expected result:

S
v
p
=
[
v
x
0
0
0
0
v
y
0
0
0
0
v
z
0
0
0
0
1
]
[
p
x
p
y
p
z
1
]
=
[
v
x
p
x
v
y
p
y
v
z
p
z
1
]
.
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:

S
v
=
[
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
s
]
.
For each vector *p* = (*p*_{x}, p_{y}, p_{z}, 1) we would have

S
v
p
=
[
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
s
]
[
p
x
p
y
p
z
1
]
=
[
p
x
p
y
p
z
1
s
]
which would be homogenized to

[
s
p
x
s
p
y
s
p
z
1
]
.
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 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. With software like RETAS UbiArt Framework and Adobe After Effects, coloring and compositing can be done in less time.

Various approaches have been developed 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.