2D computer graphics

2D computer graphics involves creating digital images on a computer, primarily from two-dimensional sources like geometric shapes, text, and existing digital pictures, using methods designed for those sources. The term can describe either the computer science field that covers these methods or the models themselves. These graphics are common in applications that grew out of older printing and drawing practices, including typography, mapmaking, technical diagrams, and advertising. In these uses, the two-dimensional image is more than a copy of a real object; it’s a standalone item with its own meaning.

Direct painting

Two-dimensional models are often chosen over 3D graphics because they offer more direct control over the image, whereas 3D graphics work more like photography than typography. In fields like desktop publishing, engineering, and business, a document described with 2D graphics techniques can be much smaller than the equivalent digital image—sometimes less than a thousandth of the size. This approach is also more adaptable, as it can be rendered at different resolutions for different output devices.

Because of this, documents and illustrations are frequently stored or sent as 2D graphic files. The field began in the 1950s with vector graphics devices, which were later mostly replaced by raster-based devices. Key milestones include the PostScript language and the X Window System protocol. 2D graphics models can combine vector graphics (geometric models), raster graphics (digital images), text for typesetting (defined by content, font, style, size, color, position, and orientation), mathematical functions and equations, and other elements.

Background (geometry)

These parts can be changed using two-dimensional geometric transformations like translation, rotation, and scaling. In object-oriented graphics, an image is described indirectly by an object with its own rendering method—a procedure that uses an arbitrary algorithm to assign colors to pixels. Complex models can be built by combining simpler objects, following object-oriented programming principles. In Euclidean geometry, a translation moves every point by a constant distance in a fixed direction.

It’s a rigid motion, along with rotations and reflections. A translation can also be seen as adding a constant vector to every point or shifting the coordinate system’s origin. A translation operator \( T_{\mathbf{\delta}} \) is defined so that \( T_{\mathbf{\delta}} f(\mathbf{v}) = f(\mathbf{v} + \mathbf{\delta}) \). If v is a fixed vector, the translation \( T_v \) works as \( T_v(p) = p + v \).

If T is a translation, the image of a subset A under T is the translation of A by T, often written as A + v. In a Euclidean space, every translation is an isometry. The set of all translations forms the translation group T, which is isomorphic to the space itself and is a normal subgroup of the Euclidean group E(n).

Translation

The quotient group E(n) / 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 one, homogeneous coordinates are typically used to represent the translation operator as a matrix, making it linear. A 3-dimensional vector w = (w_x, w_y, w_z) is written in homogeneous coordinates as w = (w_x, w_y, w_z, 1). To translate an object by a vector v, each homogeneous vector p is multiplied by this translation matrix: \[ T_{\mathbf{v}} = \begin{bmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \] As shown, the multiplication yields the translated result.