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What is computer graphics and what does it entails?
Computer graphics is one branch in computer science related to how images are formed and generated using computers. However, it is usually classified as its own division
A computer graphics process that generates an image from a model is termed "image generation". The term "generate" means generating new images based on the existing one or ones. This includes three-dimensional and two-dimensional modeling of physical objects and their surfaces, as well as animations or simulations of physical phenomena.
One model that is very popular to generate computer images is a polygon mesh, because it can easily be manipulated to fit many shapes and detailed information about the shape of the surface can be stored in vertex points and texture mapping. Image generation based on models has become common in other fields, such as visual effects for cinema and television, the motion picture industry, as well as video games.
Computer graphics also has a lot of overlap with computer-aided design, computer-aided engineering (CAD/CAE), geographic information systems, and medical image processing. The use of computers to aid in the creation, modification, or repair of objects has been present since the 1960s, and computer graphics is often distinguished from CAD as a separate field. However there has always been some overlap between the two fields (the creation of stereoscopic images in CAD or interactive 3D graphics for fly-through movies are examples of this), and much of modern three-dimensional printing technology relies on CAD software to generate and process three-dimensional objects.
Pure computer graphics work is the result of one or more 3D models that are rendered into a 2D image at high resolution and then stored in a picture file for use in various projects such as documentation, video, web design, presentation, etc. Objects can be combined mathematically to create new objects or renderings can be stylised and abstracted to create a pictorial representation of the real world.
Computer animation refers to dynamic images that are generated from a model with movement (such as animated graphics in an advertisement or computer-generated imagery for movie special effects). The term "animation" encompasses both classic animation techniques as well as modern computer generated animation, including 3D animation. Computer-generated imagery is a technique for creating moving images via the use of computer graphics software and hardware to generate three-dimensional images from models that have been created using either traditional animation techniques or 3D modeling tools.
Imaging tools are used in computer graphics to represent objects or functions. Tools commonly manipulate geometric data that has been stored in a two-dimensional array of some kind, where each entry in the array represents one coordinate of a point. Functions can be applied to the individual entries and arranged into tool sets for different purposes.
The simplest imaging tools are based on linear interpolation, such as alpha blending and the classical tool compositing. Alpha blending is important in painting software because it allows correct rendering of transparency effects that cannot be reproduced with other techniques—such as overlapping colors, or alteration of transparencies.
In computer graphics, a texture is an image applied to an object. Textures are used in both procedural and bitmap modeling methods. Textures can be stored in any image file format, including jpg, tif, png and more specialized 3d formats such as .bmp or .3ds (for Microsoft DirectX).
Procedural modeling in computer graphics
In computer graphics, procedural modeling is a category of modeling that uses algorithms for defining 3D models from a set of procedural instructions. Procedural models may be defined by mathematical equations and interpolation schemes, or by means of computer programs written in templated programming languages like Cg and HLSL. A model so produced might require further editing via polygon reduction or subdivision surface techniques. The final product of this process is called a "mesh".
The term "procedural" has also come into use as shorthand for the field of mathematics dedicated to the study of how to mathematically define 3D shapes procedurally, i.e., algorithmically.
Most recently, procedural modeling has become a major component of the emerging field of scientific visualization, in which high-quality images from large datasets are required for analysis and interpretation.
In computer graphics, polygonal modeling is a category of 3D modeling that uses polygons as its basic primitive shape. Polygons may be concave or convex (in some cases both), with the former being described as "self-intersecting" to reflect their non-planar nature and closer association with the meshes produced by subdivision surfaces. The following work provides much more detail on this subject:
Koebe's algorithm for creating minimal surfaces for computer graphics
This article presents one family of algorithms for creating minimal surfaces based on Koebe's algorithm (Koebe, 1961) for creating minimal surfaces in 3D. The basic idea behind these algorithms is to compute the real form of the polygonal surface by using Koebe's algorithm and then convert it into a minimal surface or into any other surface of lower genus. Two different real forms — biclosed and open — are obtained as a result. It is shown that if segmentation lines are removed from both real forms they coincide with each other; therefore, it is not difficult to use one real form for converting any 3D polygonal surfaces into minimal surfaces. Results of this article are applicable in many fields: computer graphics, CAD systems, engineering design, etc."
A projective representation of a NURBS surface in computer graphics is a linear subspace of the space of all possible 3D representations of a NURBS surface. In other words, there is no distinction between different points on a projective representation; they are all equally valid and indistinguishable. A point on a NURBS curve may have many different projective representations, but only one or two will be used in any given graphics system (usually either the 2D polynomial or the 3D polynomial).
Traditional computer graphics deals with 3D primitives such as points, lines and polygons (triangles or quadrilaterals) that are used to model objects. This is done to mimic the way real objects work in nature. The details at any one point can be captured by drawing many small triangles. In fact, it is possible to capture very fine detail by using just a few large triangles instead of lots of small ones which gives rise to an abruptly visible angular distortion when the level of detail changes too much from one area to another. A triangle mesh is an example of this type of representation; there are no smooth surfaces as might be found on curved objects like cars, but they do provide a good approximation when viewed from far away.
In contrast to this polygonal modeling method, NURBS models can represent any closed and bounded surface that is possibly non-planar in a single continuous mathematical definition. This is done by storing control points (vector fields) for each corner of the surface rather than using a potentially large number of vertices at each point on the surface as in polygonal meshes.
Computer graphics softwares that use NURBS surfaces are:
- CATIA 3,4,5,
- NHI (New High-Resolution),
- Parthenon3D,
- Blender3D,
- Autodesk
- Maya,
- AutoCAD,
- Visual Nastran,
- Autodesk Inventor.
Is there any good software for functions in a plane?
There are the following free computer graphics softwares that will allow you to work with algebraic functions :
- Paraview.org,
- VisIl,
- Octave3D
These programs will let you do many things. For example, you can display 3D surfaces of 3 variables functions (x,y,z) or 2 variables x-functions(x,y) or y-functions(x), z-functions(x). You can also plot function graphs of curves like polynomials and products of polynomials. Some of them even have implemented several useful commands for creating surface models (in particular Octave3D), and can export results to POV-Ray and other programs.
What is Symmetry of functions in two variables?
To understand the symmetry properties of a function we first must define what exactly we mean by the symmetry.
Symmetry means invariance under certain transforms. We need to specify the transformation in order to detect the symmetry properties. Note that all symmetries mentioned here are linear transformations, i.e they preserve lengths and angles between vectors etc.
For example, rotation by an angle of 45° around the origin is a linear transformation.
Symmetry properties apply only to real functions and not to their complex conjugates. Are there any special symmetry properties for plane curves? For every function we will find two types of symmetric parts: invariant parts and complementary parts.
How can you determine which part of a function is symmetric?
As we already noted, if we have f(x,y), then it means that f(x + ct) = f(x) where c < 0 or 0 < c < 1. The constant c in this relation depends on the forms of x and y variables:
a) If they are both quadratic, then c = 0, since f(x +ct) is linear in t.
b) If only x,y is quadratic and z variable is arbitrary (its components are not equal to zero), then c < 0 and f(x + ct) is a parabola.
c ) If only y variable is quadratic and the rest are arbitrary (their components are not equal to zero), then c > 0 and f(x+ct) has a hyperbolic shape.
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