Analytic Geometry - Three-dimensional Coordinate Systems And Beyond

Geometric figures such as points, lines, and conics are two-dimensional because they are confined to a single plane. The term two-dimensional is used because each point in this plane is represented by two real numbers. Other geometric shapes like spheres and cubes do not exist in a single plane. These shapes, called surfaces, require a third dimension to describe their location in space. To create this needed dimension, a third axis (traditionally called the z-axis) is added to the coordinate system. Consequently, the location of each point is defined by three real numbers instead of two. For example, a point defined by the coordinates (2,3,4) would be located 2 units away from the x axis, 3 units from the y axis, and 4 units from the z axis.

The algebraic equations for three-dimensional figures are determined in a way similar to their two-dimensional counterparts. For example, the equation for a sphere is x² + y² + z² = r². As can be seen, this is slightly more complicated than the equation for its two-dimensional cousin, the circle, because of the additional variable z².

It is interesting to note that just as the creation of a third dimension was possible, more dimensions can be added to our coordinate system. Mathematically, these dimensions can exist, and valid equations have been developed to describe figures in these dimensions. However, it should be noted that this does not mean that these figures physically exist and in fact, at present they only exist in the minds of people who study this type of multidimensional analytic geometry.