## Computer graphics in mathematics

Computer graphics in mathematics, specifically 2D graphics, involves the use of mathematical concepts and techniques to create and manipulate visual representations of objects and scenes in two-dimensional space on a computer screen. This field is crucial in various applications, including video games, image editing software, data visualization, computer-aided design (CAD), and more. Here are some key mathematical concepts and techniques used in 2D computer graphics:

**Cartesian Coordinates:**The fundamental basis for 2D graphics is the Cartesian coordinate system. It uses two perpendicular axes (x and y) to represent points in a 2D space. Points are denoted as (x, y), where x represents the horizontal position, and y represents the vertical position.

**Vectors and Matrices:**Vectors and matrices are essential tools for representing and transforming 2D objects. Vectors can represent points, directions, and scaling factors, while matrices are used for transformations like translation, rotation, scaling, and shearing.

**Geometric Primitives:**Basic geometric shapes such as points, lines, curves, and polygons are defined mathematically and used to create more complex 2D objects. Algorithms exist for drawing these primitives efficiently.

**Transformations:**Transformations are mathematical operations that modify the position, orientation, or size of objects in a 2D scene. Common transformations include translation (shifting), rotation (changing orientation), scaling (resizing), and reflection (flipping).

**Coordinate Systems**: Different coordinate systems can be used for various purposes, such as screen coordinates (pixel-based), world coordinates (abstract coordinates), and normalized device coordinates (a standard coordinate system used for rendering). Converting between these coordinate systems requires mathematical transformations.

**Clipping:**Clipping is the process of determining which parts of objects are visible on the screen and which are not. Various algorithms, such as Cohen-Sutherland and Liang-Barsky, use mathematical techniques to perform clipping efficiently.

**Rasterization**: Rasterization is the process of converting geometric primitives (lines, polygons) into pixels on the screen. This involves determining which pixels are inside the shape defined by the primitive and assigning colors to those pixels.

**Anti-aliasing:**Anti-aliasing techniques use mathematical methods to reduce visual artifacts like aliasing (jagged edges) in rendered images. This is achieved by blending colors at the edges of objects to create smoother transitions.

**Curves and Splines**: Curves and splines are used to create smooth, non-linear shapes in 2D graphics. Bezier curves and B-splines are examples of mathematical representations for these curves.

**Color Models:**Mathematical color models like RGB (Red, Green, Blue) and HSV (Hue, Saturation, Value) are used to represent and manipulate colors in 2D graphics. Color interpolation and blending also involve mathematical operations.

**Raster Operations (ROPs):**ROPs are bitwise operations used to combine the color values of pixels during rendering. These operations include AND, OR, XOR, and NOT, and they play a role in achieving various visual effects.

**Geometry Algorithms:**Algorithms for collision detection, hit testing, and spatial sorting are essential for interactive 2D graphics applications, such as games.

Mathematics is the foundation of 2D computer graphics, enabling the precise representation, manipulation, and rendering of visual elements. Understanding these mathematical concepts and techniques is vital for developing efficient and visually appealing 2D graphics applications.The main reason graphical objects are described by a collection of straight line segments is that the standard transformation in computer graphics map line segment onto other line segments.

Computer-aided design (CAD)is an integral part of many engineering processes such as the aircraft design process described.

## Most interactive computer software for business and industry makes use of computer graphics in the screen display and for other functions such as a graphical display of data desktop publishing and slide production for commercial and educational presentations. Consequently, anyone studying a computer language invariably spends time learning how to use at least two-dimensional (2D) graphics.

The basic mathematics used to manipulate and display a graphical image such as a wireframe model of an airplane. Sex and image consist of a number of coins, connecting lines for cobs, and information about how to fill in close regions bounded by the lines and COD lines approximated by short straight line code lines of approximated by short straight line segments and it is defined mathematically by a list of points. Among the simplest 2D graphics symbols are letters used for labels on the screen. Some letters are stored as the right frame of objects others that have COD portions are stored with additional mathematical formulas for the cal formula for the curves.

## Composite transformation

The moment of a figure on a computer screen off and requires two or more basic transformations. the composition of 6 transformations corresponds to matrix multiplication when homogenous coordinates are used.

### 3D computer graphics

In three-dimensional graphics, a biologist can examine or simulate protein molecules and search for active sites that might accept drug molecules. the biologist can rotate and translate an experimental drug and attempt to attach it to the protein. this ability to visualise Pro X shall chemical reaction is visual to modern drop and Cancer research.

Homogenous 3D coordinates

We say that (x,y,z1) homogenous coordinates for the point(x,y,z) in R^3.

In general (X, Y, Z, H) are homogenous coordinates for(x,y,z) ifH will not equal zero

X=X/H

y=Y/H

z=Z/H.

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