Next on the agenda is, therefore, the transformations to make the 3D objects move. The first of these is very simple: if you walk towards an objects, or back away, what happens? Well, the objects becomes larger or smaller, but the perspective deals with that; all we're worried about is the polygons' z-values. If z increases into the monitor, then if you walk towards some polygons, their z values must be decreased as you walk towards them; and increased if you back away from them. We assume that polygons with negative z values are behind you, and so not visible. That's the easy part; if you want to turn yourself, the whole 3D world must be rotated about the place where you are standing. This is a tricky business to work out, but the secret is using matrix transformation (this used to be in the GCSE maths course - seriously). If your're unfamiliar with these it works like this. A polygon is stored as a matrix of 3 rows (x,y and z) and 4 columbs, one for each vertex. If this matrix is multiplied by the correct transformation matrix, then the polygon is effectively rotated about the origin. Matrix multiplication is not as easy as it sounds: effectively, each product is the sum of the products of a columb times a row. The matrix on the left represents one polygon; the columbs contain the x, y and z coords of each vertex respectivly. The matrix on the right is the transformation matrix: more on that later. To generate the first columb of the first row, the first row must be multiplied by the first columb (always along the row first): that is, each element (a,b,c) is multiplied by the corresponding element (1,2,3) in the transformation matrix, and the results summed, to give give the result, A. This result is the x coordinate of the first point of the transformed polygon. All 12 entries are calculated in this way: and the result is a polygon rotated about the origin. Critical to this process, of course, is the transformation matrix.