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# 4.1: Introduction

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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.