Three-dimensional Co-ordinate Geometry

Abstract

Before we lead on to a study of the graphical display of objects in three-dimensional space, we first have to come to terms with the three-dimensional Cartesian co-ordinate geometry and introduce some useful procedures for manipulating objects in three-dimensional space. (For further reading we recommend books by Cohn (1961) and McCrae (1953)). As in two-dimensional space, we arbitrarily fix a point in the space, named the co-ordinate origin (origin for short). We then imagine three mutually perpendicular lines through this point, each line extending to infinity in both directions. These are the x-axis, y-axis and z-axis. Each axis is thought to have a positive and a negative half, both starting at the origin — that is, distances measured from the origin along the axis are positive on one side and negative on the other. We may think of the x and y axes in a similar way to two-dimensional space, both lying on the page of this book say, the positive x-axis horizontal and to the right of the origin, and the positive y-axis vertical and above the origin. This just leaves the position of the z-axis: it has to be perpendicular to the page (since it is perpendicular to both x and y axes). The positive z-axis can be into the page (the so-called left-handed triad of axes) or out of the page (the right-handed triad). You can realise the difference on your hands. On either hand, hold the thumb, index finger and middle finger at right angles to one another with the middle finger perpendicular to the palm of your hand: the thumb may be taken as the positive x-axis, the index finger as the positive y-axis and the middle finger the positive z-axis. See figure 6.1.