Copyright (c) Susan Laflin. August 1999.

In the following sections, we shall discuss the topics as though we were dealing with idealised mathematical objects, points with position but no size and lines and planes of zero thickness. Obviously this does not correspond to the real world where even the thinnest plane is hundreds of atoms in thickness. However the ideas can be developed without bothering about the effects of thickness, the need to specify whether we are discussing the centre of the line or one of its outer edges, and these extra complications can be considered later.

When we come to draw the resulting diagrams on paper or a computer screen, we shall have to move from the mathematical ideal to a pattern of lines of known thickness or of pixels on a screen which may be interpreted by those looking at them as a representation of the mathematical ideal. In addition, we are attempting to represent the idea of three-dimensional objects by a pattern of lines or dots in two-dimensions. There are certain well-known techniques (you can call them tricks if you are feeling unkind) which fool the human eye into imagining a solid three-dimensional object. These will also be discussed in this section.

When dealing with three-dimensional graphics, three coordinates are needed to specify a point uniquely. These are usually the coordinates (x,y,z) relating to a set of Cartesian coordinates, but this is not essential. For example, polar coordinates may be used and values quoted for latitude, longitude and radius. These will give a unique position for each point and once again three values are needed to specify it.

Right and Left Handed Axes

The three-dimensional Cartesian coordinates may be either right-handed or left-handed. To visualise this, you should hold up your right or left hand, with the thumb and first two fingers at right angles to each other and this will demonstrate the direction of these axes.

When we come to use these coordinates on a computer terminal, most systems still use the two-dimensional version with the origin in the bottom left-hand corner of the screen, the x-axis running from left to right along the bottom of the screen and the y-axis from bottom to top up the left-hand side of the screen. A right-handed set of axes then has the z-axis coming out of the screen towards the user and a left-handed set has the z-axis going into the screen away from the user.

Most software then has some means of reducing the three-dimensional object to a two-dimensional drawing in order to view the object. Possible means are to ignore the z-value, thus giving a parallel or orthogonal projection onto the screen, to calculate a perspective projection onto the screen or to take a slice through the object in the plane of the screen. The projections may or may not have any provision for hidden-line or hidden-surface removal.