4.1: Introduction

Various geometrical figures in three-dimensional space can be described relative to a set of mutually orthogonal axes O$x$, O$y$, O$z$, and a point can be represented by a set of rectangular coordinates $(x, y, z)$. The point can also be represented by cylindrical coordinates $( ρ , \phi , z)$ or spherical coordinates $(r , θ , \phi )$, which were described in Chapter 3. In this chapter, we are concerned mostly with $(x, y, z)$. The rectangular axes are usually chosen so that when you look down the $z$-axis towards the $xy$-plane, the $y$-axis is $90^\circ$ counterclockwise from the $x$-axis. Such a set is called a right-handed set. A left-handed set is possible, and may be useful under some circumstances, but, unless stated otherwise, it is assumed that the axes chosen in this chapter are right-handed.

An Equation connecting $x$, $y$ and $z$, such as

$$f(x,y,z) = 0 \label{4.1.1} \tag{4.1.1}$$

or $$z = z(x,y) \label{4.1.2} \tag{4.1.2}$$

describes a two-dimensional surface in three-dimensional space. A line (which need be neither straight nor two-dimensional) can be described as the intersection of two surfaces, and hence a line or curve in three-dimensional coordinate geometry is described by two Equations, such as

$$f(x,y,z) = 0 \label{4.1.3} \tag{4.1.3}$$

and $$g (x,y,z) = 0 . \label{4.1.4} \tag{4.1.4}$$

In two-dimensional geometry, a single Equation describes some sort of a plane curve. For example,

$$y^2 = 4qx \label{4.1.5} \tag{4.1.5}$$

describes a parabola. But a plane curve can also be described in parametric form by two Equations. Thus, a parabola can also be described by

$$x = qt^2 \label{4.1.6} \tag{4.1.6}$$

and $$y = 2qt \label{4.1.7} \tag{4.1.7}$$

Similarly, in three-dimensional geometry, a line or curve can be described by three Equations in parametric form. For example, the three Equations

$$x = a \cos t \label{4.1.8} \tag{4.1.8}$$

$$y = a \sin t \label{4.1.9} \tag{4.1.9}$$

$$z = ct \label{4.1.10} \tag{4.1.10}$$

describe a curve in three-space. Think of the parameter $t$ as time, and see if you can imagine what sort of a curve this is.

We shall be concerned in this chapter mainly with six types of surface: the plane, the ellipsoid, the paraboloid, the hyperboloid, the cylinder and the cone.