It is of critical importance to know how to measure distances between points in space. The formula for doing so is based on measuring distance in the plane, and is known (in both contexts) as the Euclidean measure of distance.

We refer to the line segment that connects points \(P\) and \(Q\) in space as \(\overline{PQ}\text{,}\) and refer to the length of this segment as \(\norm{\overline{PQ}}\text{.}\) The above distance formula allows us to compute the length of this segment.

Just as a circle is the set of all points in the plane equidistant from a given point (its center), a sphere is the set of all points in space that are equidistant from a given point. Definition 10.1.3 allows us to write an equation of the sphere.

We start with a point \(C = (a,b,c)\) which is to be the center of a sphere with radius \(r\text{.}\) If a point \(P=(x,y,z)\) lies on the sphere, then \(P\) is \(r\) units from \(C\text{;}\) that is, \begin{equation*} \norm{\overline{PC}} = \sqrt{(x-a)^2+(y-b)^2+(z-c)^2} = r. \end{equation*}

Squaring both sides, we get the standard equation of a sphere in space with center at \(C=(a,b,c)\) with radius \(r\text{,}\) as given in the following Key Idea.

The equation of a sphere is an example of an implicit function defining a surface in space. In the case of a sphere, the variables \(x\text{,}\) \(y\) and \(z\) are all used. We now consider situations where surfaces are defined where one or two of these variables are absent.

The coordinate axes naturally define three planes (shown in Figure 10.1.3), the coordinate planes: the \(x\)-\(y\) plane, the \(y\)-\(z\) plane and the \(x\)-\(z\) plane. The \(x\)-\(y\) plane is characterized as the set of all points in space where the \(z\)-value is 0. This, in fact, gives us an equation that describes this plane: \(z=0\text{.}\) Likewise, the \(x\)-\(z\) plane is all points where the \(y\)-value is 0, characterized by \(y=0\text{.}\)

The equation \(x=2\) describes all points in space where the \(x\)-value is 2. This is a plane, parallel to the \(y\)-\(z\) coordinate plane, shown in Figure 10.1.9.

The equation \(x=1\) obviously lacks the \(y\) and \(z\) variables, meaning it defines points where the \(y\) and \(z\) coordinates can take on any value. Now consider the equation \(x^2+y^2=1\) in space. In the plane, this equation describes a circle of radius 1, centered at the origin. In space, the \(z\) coordinate is not specified, meaning it can take on any value. In Figure 10.1.12 (a), we show part of the graph of the equation \(x^2+y^2=1\) by sketching 3 circles: the bottom one has a constant \(z\)-value of \(-1.5\text{,}\) the middle one has a \(z\)-value of 0 and the top circle has a \(z\)-value of 1. By plotting all possible \(z\)-values, we get the surface shown in Figure 10.1.12 (b). This surface looks like a “tube,” or a “cylinder”; mathematicians call this surface a cylinder for an entirely different reason.

In this text, we consider curves \(C\) that lie in planes parallel to one of the coordinate planes, and lines \(L\) that are perpendicular to these planes, forming right cylinders. Thus the directrix can be defined using equations involving 2 variables, and the rulings will be parallel to the axis of the 3\(^\text{ rd }\) variable.

In the example preceding the definition, the curve \(x^2+y^2=1\) in the \(x\)-\(y\) plane is the directrix and the rulings are lines parallel to the \(z\)-axis. (Any circle shown in Figure 10.1.12 can be considered a directrix; we simply choose the one where \(z=0\text{.}\)) Sample rulings can also be viewed in part (b) of the figure. More examples will help us understand this definition.

One of the applications of integration we learned previously was to find the volume of solids of revolution — solids formed by revolving a curve about a horizontal or vertical axis. We now consider how to find the equation of the surface of such a solid.

Consider the surface formed by revolving \(y=\sqrt{x}\) about the \(x\)-axis. Cross–sections of this surface parallel to the \(y\)-\(z\) plane are circles, as shown in Figure 10.1.19(a). Each circle has equation of the form \(y^2+z^2=r^2\) for some radius \(r\text{.}\) The radius is a function of \(x\text{;}\) in fact, it is \(r(x) = \sqrt{x}\text{.}\) Thus the equation of the surface shown in Figure 10.1.19b is \(y^2+z^2=(\sqrt{x})^2\text{.}\)

We generalize the above principles to give the equations of surfaces formed by revolving curves about the coordinate axes.

This particular method of creating surfaces of revolution is limited. For instance, in Example 7.3.18 of Section 7.3 we found the volume of the solid formed by revolving \(y=\sin(x)\) about the \(y\)-axis. Our current method of forming surfaces can only rotate \(y=\sin(x)\) about the \(x\)-axis. Trying to rewrite \(y=\sin(x)\) as a function of \(y\) is not trivial, as simply writing \(x=\sin^{-1}(y)\) only gives part of the region we desire.

What we desire is a way of writing the surface of revolution formed by rotating \(y=f(x)\) about the \(y\)-axis. We start by first recognizing this surface is the same as revolving \(z=f(x)\) about the \(z\)-axis. This will give us a more natural way of viewing the surface.

A value of \(x\) is a measurement of distance from the \(z\)-axis. At the distance \(r\text{,}\) we plot a \(z\)-height of \(f(r)\text{.}\) When rotating \(f(x)\) about the \(z\)-axis, we want all points a distance of \(r\) from the \(z\)-axis in the \(x\)-\(y\) plane to have a \(z\)-height of \(f(r)\text{.}\) All such points satisfy the equation \(r^2=x^2+y^2\text{;}\) hence \(r=\sqrt{x^2+y^2}\text{.}\) Replacing \(r\) with \(\sqrt{x^2+y^2}\) in \(f(r)\) gives \(z=f(\sqrt{x^2+y^2})\text{.}\) This is the equation of the surface.

Spheres, planes and cylinders are important surfaces to understand. We now consider one last type of surface, a quadric surface. The definition may look intimidating, but we will show how to analyze these surfaces in an illuminating way.

When the coefficients \(D\text{,}\) \(E\) or \(F\) are not zero, the basic shapes of the quadric surfaces are rotated in space. We will focus on quadric surfaces where these coeffiecients are 0; we will not consider rotations. There are six basic quadric surfaces: the elliptic paraboloid, elliptic cone, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, and the hyperbolic paraboloid.

We study each shape by considering traces, that is, intersections of each surface with a plane parallel to a coordinate plane. For instance, consider the elliptic paraboloid \(z= x^2/4+y^2\text{,}\) shown in Figure 10.1.33. If we intersect this shape with the plane \(z=d\)  (i.e., replace \(z\) with \(d\)), we have the equation: \begin{align*} d \amp = \frac{x^2}4+y^2.\\ \end{align*} Divide both sides by \(d\text{:}\) \begin{align*} 1 \amp = \frac{x^2}{4d} + \frac{y^2}{d}. \end{align*}

This describes an ellipse — so cross sections parallel to the \(x\)-\(y\) coordinate plane are ellipses. This ellipse is drawn in the figure.

Now consider cross sections parallel to the \(x\)-\(z\) plane. For instance, letting \(y=0\) gives the equation \(z=x^2/4\text{,}\) clearly a parabola. Intersecting with the plane \(x=0\) gives a cross section defined by \(z=y^2\text{,}\) another parabola. These parabolas are also sketched in the figure.

Thus we see where the elliptic paraboloid gets its name: some cross sections are ellipses, and others are parabolas.

Such an analysis can be made with each of the quadric surfaces. We give a sample equation of each, provide a sketch with representative traces, and describe these traces.

Elliptic Paraboloid, \(\ds z=\frac{x^2}{a^2}+\frac{y^2}{b^2}\)

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Elliptic Cone, \(\ds z^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}\)

Ellipsoid, \(\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)

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Hyperboloid of One Sheet, \(\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)

Hyperboloid of Two Sheets, \(\ds \frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)

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Hyperbolic Paraboloid, \(\ds z=\frac{x^2}{a^2}-\frac{y^2}{b^2}\)

This section has introduced points in space and shown how equations can describe surfaces. The next sections explore vectors, an important mathematical object that we'll use to explore curves in space.