Graphics

Mapping 2D and 3D Coordinate Systems

Rene' Descartes, (1596 - 1690), French philosopher, mathematician

To find the approximate or exact location of something, at least two things must be known,... where you are, and a unit of measure to calculate distance.

If only these two are known, you can fairly accurately state the distance you are from the measured point. To precisely locate the point within a three dimensional space it is necessary to know one other point's distance from you within the space.

This is the basis of celestial navigation, and in fact all types of mapping (or cartography as it is called). 'Triangulation' is a method for finding your location when you can see two other locations.

For instance, in a boat... You are at sea at night and can see the stars... you could calculate your position in logitude and latitude by finding two stars on a celestial navigation chart, and measuring the angles of the stars to your position, (using a sextant), you could 'triangulate' your relative position.

Much of what has to do with displaying computer graphics comes from this type of 'technology'.

In the 1600's, Rene' Descartes, (1596 - 1690), a French philosopher and mathematician detailed what is known as the 'Cartesian Coordinate System'. Descartes, for whom 'cartography' is named, theorized that given a unit of measure and reference points which form a virtual grid, you could accurately map anything.

Two Dimensional Space In a two-dimensional cartesian space, distances are referenced in units of the whole where the X axis spans the width, and the y axis spans the height, (width and height are relative terms, the system applies to any two dimensions).

Three Dimensional Space In a three dimensional coordinate system, X and Y axis denote the width and height, and the third dimension, "depth", is plotted on the Z axis. For the most part, computer graphics are plotted in relation to where they will appear on the display screen or printout. Since the display is a two dimensional view, depth is only relevant for rendering or modeling of 3D objects.

     

Using the Cartesian Coordinate system to describe or render images is fairly straightforward, although using an abstract mathematical process may at first seem awkward.

Device Independent Graphics Plotting and Rendering

     

A basic understanding of vector graphics will help.   Understanding relative coordinates is also helpful.   The nicest thing about using this method of vectors and relative coordinates allows for device independence, that is to say that the display screen or print area for the desired plot may be any size.

The abstract concept requires that you consider the overall area of the design to be "1 unit high" and "1 unit wide" (and in the case of 3D plots, to also be "1 unit deep").

To plot a point which is exactly in the middle of the display or print area simply define the point as being at (.5)x, (.5)y, or at one half the x axis and one half the y axis.

To describe a line which runs diagonal from one corner of the display to the other, you would define the line using two coordinate pairs, (0x, 0y) - (1x, 1y).  This definition worked regardless of the size of the area and demonstrates how scalable the Cartesian coordinate system is.

More complex shapes, characters, fonts, and other design elements can be plotted using the same methods.  Using lists of coordinate pairs, complex polygons, irregular lines and other shapes may be plotted and rendered on any device.