HA4 – Geometric Theory

Geometry - Cartesian Coordinates System

Cartesian coordinates, also called rectangular coordinates, provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface or in three-dimensional ( 3D ) space. The scheme gets its name from one of the first people known to have used it, the French mathematician and philosopher René Descartes (1596-1650). The Cartesian coordinate system is used to define positions on computer displays and in virtual reality (VR) renderings. The system is also employed in mathematics, physics, engineering, navigation, robotics, economics, and other sciences.

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Vector

A vector is a quantity or phenomenon that has t...

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Co-ordinates

Coordinates are distances or angles.

The Cartesian plane consists of two perpendicular axes that cross at a central point called the origin. Positions or coordinates are determined according to the east/west and north/south displacements from the origin. The east/west axis is often called the x axis, and the north/south axis is called the y axis. For this reason, the Cartesian plane is also known as the x y -plane. The x and y axes are linear number lines, meaning that each division on a given axis always represents the same increment. However, the increments on different axes can differ. For example, in the illustration at left below, each increment on the x axis might represent 2 units, while each increment on the y axis represents 5 units. Points or coordinates are indicated by writing an opening parenthesis, the x value, a comma, the y value, and a closing parenthesis in that order. An example is (x, y) = (2,-5). The origin is usually, but not always, assigned the value (0,0).

Cartesian three-space, also called xyz -space, has a third axis, oriented at right angles to the x y -plane. This axis, usually called the z axis, passes through the origin of the xy -plane. Positions or coordinates are determined according to the east/west (x), north/south ( y ), and up/down ( z ) displacements from the origin. As is the case with the x and y axes, the z axis is a linear number line. For example, in the illustration at right above, each increment on the x axis might represent 5 units, each increment on the y axis 10 units, and each increment on the z axis 2 units. Points or coordinates are indicated by writing an opening parenthesis, the x value, a comma, the y value, another comma, the z value, and a closing parenthesis in that order. An example is ( x, y, z ) = (5,-10,-4). The origin is usually, but not always, assigned the value (0, 0, 0).

http://www.basic-mathematics.com/cartesian-coordinate-system.html

http://whatis.techtarget.com/definition/Cartesian-coordinates-rectangular-coordinates

Polygon in video games

- Wireframes are an important design tool used in Web development. It is a visualization tool for presenting proposed functions, structure and content of a Web page or Web site. A wireframe separates the graphic elements of a Web site from the functional elements in such a way that Web teams can easily explain how users will interact with the Web site. A typical wireframe includes:

(1) key page elements and their location, such as header, footer, navigation, content objects, branding elements, (2) grouping of elements, such as side bars, navigation bars, content areas, (3) labelling, page title, navigation links, headings to content objects, and (4) place holders, content text and images. http://www.webopedia.com/TERM/W/wireframe.html

(2) Image - http://cgcookie.com/unity/2013/04/29/modeling-and-texturing-polygon-hair-for-game-characters/

Primitives

A low-level object or operation from which higher-level, more complex objects and operations can be constructed. In graphics, primitives are basic elements, such as lines, curves, and polygons, which you can combine to create more complex graphical images. In programming, primitives are the basic operations supported by the programming language. A programmer combines these primitives to create more complex operations, which are packaged as functions, procedures, and methods.

http://www.webopedia.com/TERM/P/primitive.html

Polygons and Shading - S uface

The vast majority of 3-D objects created for computer games are made up of polygons. A polygon is an area defined by lines. To have a polygon, you must have at least three lines. The lines connect a series of coordinates in the three-dimensional "space" the computer creates. The point where the lines connect is known as a vertex. Each vertex has X, Y and Z coordinates.

•X determines the position relative to right or left in the virtual space

•Y determines the position relative to top or bottom in the virtual space

•Z determines the position relative to front or back in the virtual space

Once each polygon has a set of vertices to define its shape, it needs information that tells it what to look like. There are four common ways to do this:

•Flat shading

•Gouraud shading

•Phong shading

•Texture mapping

Flat shading simply assigns a single colour to a polygon. It is very simple and fast, but makes the object look artificial. Gouraud shading is more complex. Colours are assigned to each vertex and then are blended across the face of the polygon. Since each vertex is typically associated with at least three distinct polygons, this makes the object look natural instead of faceted.

http://www.slideshare.net/upload?from_source=loggedin_newsfeed

http://forums.eidosgames.com/showthread.php?t=108926