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Coordinate System Representation in computer graphics

Coordinate System Representation in computer graphics

Coordinate Representation

Normally, graphics packages required coordinate specifications to be given with respect to coordinate reference frames.

To describe a picture, the world-coordinate reference frame (2D or 3D) must be selected.

Rene Descartes, the French mathematician, and philosopher, discovered that some algebraic equations could be interpreted geometrically by graphing them onto a two-dimensional coordinate system. He had founded analytic geometry. The rectangular, two- dimensional system he used to graph algebraic functions bears his name Cartesian co-ordinate system.

2D Cartesian Coordinate system

Figure 1.27: 2D Cartesian Coordinate system

The two-dimensional Cartesian coordinate system has well-defined properties. To construct two-dimensional Cartesian coordinate system one has to use two (hypothetically) unbounded straight lines intersecting at right angles to form the Principal axes x and y, with positive and negative directions. indicated (ligure 1.27). Their point of intersection O is the origin. One imposes a grid of equally spaced lines parallel to and in the plane of the principal axes, which forms the basis of all measurements and analysis. Every point in the plane of uns coordinate system is defined by a pair of numbers (x, y), its coordinates.

The coordinates of p, in the figure are (X1, Y1). To locate pi in the system, we construct a line parallel to the y axis through x1, and parallel to the x axis through y1. These two lines intersect at p1.

The location and orientation of a coordinate system are arbitrary, and there may be more than one.

The polar coordinate system has limited use in computer graphics and geometric modeling, but its relationship to the Cartesian system suggests how he/she might rotate points. Figure 1.28 a illustrates a polar coordinate system:

Relationship between Cartesian and Polar Coordinate System

Figure 1.28: Relationship between Cartesian and Polar Coordinate System

Points in it are located by giving their distance r from the origin and the angular displacement of the line of r with respect to a reference line. Thus, the polar coordinates of a point are (r, ).

One can convert between polar coordinates and Cartesian coordinates. Figure 1.28 b shows the two systems superimposed so that their origins coincide and the polar reference line and x axis are collinear. The transformation equations from polar to Cartesian are derived using simple trigonometry. Thus,

x = r cos  

y =r sin

Screen Coordinates

A graphics device has a “screen” (possibly a piece of paper) of specified size and shape. Most of screens are rectangular and are described by a “pixel” coordinate system which labels the rows and columns of the screen. These row and column coordinates will be called screen, device or display coordinates.

Computer screens typically use a coordinate system which also consists of an origin and two axes, at right angles to each other, intersecting at a point called the origin.

Each point on the plane containing the coordinates is uniquely identified by two numbers:

1) The first represents the signed horizontal distance from the vertical axis, and

2) Second represents the signed vertical distance from the horizontal axis. The horizontal dimension increases to the right from the origin, and the vertical dimension downward from the origin.