In the previous section we discussed planes in Euclidean space. A plane is an example of
a*surface, which we will define informally*8 as the solution set of the equation*F*(x,*y,z)*_{=}0
in_{R}3, for some real-valued function *F*. For example, a plane given by*ax*_{+}*b y*_{+}*cz*_{+}*d*_{=}0
is the solution set of*F(x,y,z)*_{=}0 for the function*F*(x,*y,z)*_{=}*ax*_{+}*b y*_{+}*cz*_{+}*d. Surfaces are*
2-dimensional. The plane is the simplest surface, since it is “flat”. In this section we will
look at some surfaces that are more complex, the most important of which are the sphere
and the cylinder.

**Definition 1.9.** A**sphere***S*is the set of all points (x,*y,z) in*_{R}3 which are a fixed distance*r*
(called the**radius**) from a fixed point*P*0=(x0,*y*0,*z*0) (called the**center**of the sphere):

*S*_{=}{(x,*y,z) : (x*_{−}*x*0)2+(*y*−*y*0)2+(z−*z*0)2=*r*2} (1.29)
Using vector notation, this can be written in the equivalent form:

*S*_{=}{**x**:_{k}**x**_{−}**x**0k =*r*} (1.30)
where**x**_{=}(x,*y,z) and***x**0=(x0,*y*0,*z*0) are vectors.

Figure 1.6.1 illustrates the vectorial approach to spheres.

*y*
*z*
*x*
0
k**x**_{k =}*r*
**x**

**(a)**radius*r*, center (0, 0, 0)

*y*
*z*
*x*
0
k**x**_{−}**x**0k =*r*
**x**
**x**0
**x**_{−}**x**0
(x0,*y*0,*z*0)
**(b)**radius*r*, center (*x*0,*y*0,*z*0)
**Figure 1.6.1** Spheres inR3

Note in Figure 1.6.1(a) that the intersection of the sphere with the *x y-plane is a circle*
of radius *r* (i.e. a*great circle, given by* *x*2_{+}*y*2_{=}*r*2 as a subset of _{R}2). Similarly for the
intersections with the *xz-plane and the* *yz-plane. In general, a plane intersects a sphere*
either at a single point or in a circle.

**1.6 Surfaces** **41**

* Example 1.27.*Find the intersection of the sphere

*x*2

_{+}

*y*2

_{+}

*z*2

_{=}169 with the plane

*z*

_{=}12.

*y*
*z*
*x*
0
*z*_{=}12
**Figure 1.6.2**
*Solution:*The sphere is centered at the origin and has radius

13_{=}p169, so it does intersect the plane *z*_{=}12. Putting
*z*_{=}12 into the equation of the sphere gives

*x*2_{+}*y*2_{+}122_{=}169

*x*2_{+}*y*2_{=}169_{−}144_{=}25_{=}52

which is a circle of radius 5 centered at (0, 0, 12), parallel to
the*x y-plane (see Figure 1.6.2).*

If the equation in formula (1.29) is multiplied out, we get an equation of the form:

*x*2_{+}*y*2_{+}*z*2_{+}*ax*_{+}*b y*_{+}*cz*_{+}*d*_{=}0 (1.31)
for some constants*a,b,c*and*d. Conversely, an equation of this formmay*describe a sphere,
which can be determined by completing the square for the*x,* *y*and*z*variables.

* Example 1.28.*Is 2x2

_{+}2

*y*2

_{+}2z2

_{−}8x

_{+}4

*y*

_{−}16z

_{+}10

_{=}0 the equation of a sphere?

*Solution:*Dividing both sides of the equation by 2 gives

*x*2_{+}*y*2_{+}*z*2_{−}4x_{+}2*y*_{−}8z_{+}5_{=}0

(x2_{−}4x_{+}4)_{+}(y2_{+}2*y*_{+}1)_{+}(z2_{−}8z_{+}16)_{+}5_{−}4_{−}1_{−}16_{=}0
(x_{−}2)2_{+}(y_{+}1)2_{+}(z_{−}4)2_{=}16

which is a sphere of radius 4 centered at (2,_{−}1, 4).

* Example 1.29.*Find the points(s) of intersection (if any) of the sphere from Example 1.28
and the line

*x*

_{=}3

_{+}

*t,*

*y*

_{=}1

_{+}2t,

*z*

_{=}3

_{−}

*t.*

*Solution:* Put the equations of the line into the equation of the sphere, which was (x_{−}2)2_{+}
(y_{+}1)2_{+}(z_{−}4)2_{=}16, and solve for*t:*

(3_{+}*t*_{−}2)2_{+}(1_{+}2t_{+}1)2_{+}(3_{−}*t*_{−}4)2_{=}16
(t_{+}1)2_{+}(2t_{+}2)2_{+}(_{−}*t*_{−}1)2_{=}16
6t2_{+}12t_{−}10_{=}0
The quadratic formula gives the solutions *t*_{= −}1_{±}_{p}4

6. Putting those two values into the equations of the line gives the following two points of intersection:

µ
2_{+}_{p}4
6,−1+
8
p
6, 4−
4
p
6
¶
and
µ
2_{−}_{p}4
6,−1−
8
p
6, 4+
4
p
6
¶

If two spheres intersect, they do so either at a single point or in a circle.

* Example 1.30.*Find the intersection (if any) of the spheres

*x*2

_{+}

*y*2

_{+}

*z*2

_{=}25 and

*x*2

_{+}

*y*2

_{+}(z

_{−}2)2

_{=}16.

*Solution:*For any point (x,*y,z) on both spheres, we see that*
*x*2_{+}*y*2_{+}*z*2_{=}25 _{⇒} *x*2_{+}*y*2_{=}25_{−}*z*2, and
*x*2_{+}*y*2_{+}(z_{−}2)2_{=}16 _{⇒} *x*2_{+}*y*2_{=}16_{−}(z_{−}2)2, so
16_{−}(z_{−}2)2_{=}25_{−}*z*2 _{⇒} 4*z*_{−}4_{=}9 _{⇒} *z*_{=}13/4

⇒ *x*2_{+}*y*2_{=}25_{−}(13/4)2_{=}231/16
∴The intersection is the circle*x*2_{+}*y*2_{=}231_{16} of radius

p

231

4 ≈3.8 centered at (0, 0, 13

4).

The cylinders that we will consider are *right circular cylinders. These are cylinders ob-*
tained by moving a line*L*along a circle *C*inR3in a way so that *L*is always perpendicular
to the plane containing*C. We will only consider the cases where the plane containingC* is
parallel to one of the three coordinate planes (see Figure 1.6.3).

*y*
*z*
*x*
0
*r*
**(a)***x*2+*y*2=*r*2, any*z*
*y*
*z*
*x*
0
*r*
**(b)***x*2+*z*2=*r*2, any*y*
*y*
*z*
*x*
0
*r*
**(c)***y*2+*z*2=*r*2, any*x*
**Figure 1.6.3** Cylinders inR3

For example, the equation of a cylinder whose base circle *C* lies in the *x y-plane and is*
centered at (a,*b, 0) and has radius* *r*is

(x_{−}*a)*2_{+}(*y*_{−}*b)*2_{=}*r*2, (1.32)
where the value of the *z*coordinate is unrestricted. Similar equations can be written when
the base circle lies in one of the other coordinate planes. A plane intersects a right circular
cylinder in a circle, ellipse, or one or two lines, depending on whether that plane is parallel,
oblique9, or perpendicular, respectively, to the plane containing *C. The intersection of a*
surface with a plane is called the**trace**of the surface.

**1.6 Surfaces** **43**

The equations of spheres and cylinders are examples of*second-degree equations*in_{R}3, i.e.
equations of the form

*Ax*2_{+}*B y*2_{+}*C z*2_{+}*D x y*_{+}*Exz*_{+}*F yz*_{+}*G x*_{+}*H y*_{+}*I z*_{+}*J*_{=}0 (1.33)
for some constants*A,B,*. . . ,*J. If the above equation is not that of a sphere, cylinder, plane,*
line or point, then the resulting surface is called a**quadric surface**.

*y*
*z*
*x*
0
*a* *b*
*c*
**Figure 1.6.4** Ellipsoid
One type of quadric surface is the**ellipsoid**, given

by an equation of the form:
*x*2
*a*2+
*y*2
*b*2+
*z*2
*c*2=1 (1.34)
In the case where *a*_{=}*b*_{=}*c, this is just a sphere.*
In general, an ellipsoid is egg-shaped (think of an
ellipse rotated around its major axis). Its traces in
the coordinate planes are ellipses.

Two other types of quadric surfaces are the **hyperboloid of one sheet**, given by an
equation of the form:

*x*2
*a*2+
*y*2
*b*2−
*z*2
*c*2 =1 (1.35)

and the**hyperboloid of two sheets**, whose equation has the form:
*x*2
*a*2−
*y*2
*b*2−
*z*2
*c*2 =1 (1.36)
*y*
*z*
*x*
0

**Figure 1.6.5** Hyperboloid of one sheet

*y*
*z*

*x*

0

For the hyperboloid of one sheet, the trace in any plane parallel to the *x y-plane is an*
ellipse. The traces in the planes parallel to the*xz- or* *yz-planes are hyperbolas (see Figure*
1.6.5), except for the special cases*x*_{= ±}*a*and*y*_{= ±}*b; in those planes the traces are pairs of*
intersecting lines (see Exercise 8).

For the hyperboloid of two sheets, the trace in any plane parallel to the*x y- orxz-plane is*
a hyperbola (see Figure 1.6.6). There is no trace in the*yz-plane. In any plane parallel to the*
*yz-plane for which*_{|}*x*_{| > |}*a*_{|}, the trace is an ellipse.

*y*
*z*

*x*

0

**Figure 1.6.7** Paraboloid
The **elliptic paraboloid** is another type of quadric surface,

whose equation has the form:
*x*2
*a*2+
*y*2
*b*2 =
*z*
*c* (1.37)

The traces in planes parallel to the*x y-plane are ellipses, though*
in the *x y-plane itself the trace is a single point. The traces in*
planes parallel to the *xz- or* *yz-planes are parabolas. Figure*
1.6.7 shows the case where *c*_{>}0. When *c*_{<}0 the surface is
turned downward. In the case where*a*_{=}*b, the surface is called*
a*paraboloid of revolution, which is often used as a reflecting sur-*
face, e.g. in vehicle headlights.10

A more complicated quadric surface is the**hyperbolic paraboloid**, given by:
*x*2
*a*2−
*y*2
*b*2=
*z*
*c* (1.38)
-10
-5
0
5
10
-10
-5
0
5
10
-100
-50
0
50
100
*z*
*x*
*y*
*z*

**Figure 1.6.8** Hyperbolic paraboloid

**1.6 Surfaces** **45**

The hyperbolic paraboloid can be tricky to draw; using graphing software on a computer
can make it easier. For example, Figure 1.6.8 was created using the free Gnuplot package
(see Appendix C). It shows the graph of the hyperbolic paraboloid *z*_{=}*y*2_{−}*x*2, which is the
special case where*a*_{=}*b*_{=}1 and*c*_{= −}1 in equation (1.38). The mesh lines on the surface are
the traces in planes parallel to the coordinate planes. So we see that the traces in planes
parallel to the*xz-plane are parabolas pointing upward, while the traces in planes parallel*
to the *yz-plane are parabolas pointing downward. Also, notice that the traces in planes*
parallel to the *x y-plane are hyperbolas, though in the* *x y-plane itself the trace is a pair of*
intersecting lines through the origin. This is true in general when *c*_{<}0 in equation (1.38).
When *c*_{>}0, the surface would be similar to that in Figure 1.6.8, only rotated 90◦ around
the *z-axis and the nature of the traces in planes parallel to the* *xz- or* *yz-planes would be*
reversed.

*y*
*z*

*x*

0

**Figure 1.6.9** Elliptic cone
The last type of quadric surface that we will consider is the

**elliptic cone**, which has an equation of the form:
*x*2
*a*2+
*y*2
*b*2−
*z*2
*c*2 =0 (1.39)

The traces in planes parallel to the *x y-plane are ellipses, ex-*
cept in the *x y-plane itself where the trace is a single point.*
The traces in planes parallel to the*xz- oryz-planes are hyper-*
bolas, except in the *xz- and* *yz-planes themselves where the*
traces are pairs of intersecting lines.

Notice that every point on the elliptic cone is on a line which
lies entirely on the surface; in Figure 1.6.9 these lines all go
through the origin. This makes the elliptic cone an example of
a*ruled surface. The cylinder is also a ruled surface.*

What may not be as obvious is that both the hyperboloid of one sheet and the hyperbolic
paraboloid are ruled surfaces. In fact, on both surfaces there are *two* lines through each
point on the surface (see Exercises 11-12). Such surfaces are called *doubly ruled surfaces,*
and the pairs of lines are called a*regulus.*

It is clear that for each of the six types of quadric surfaces that we discussed, the surface
can be translated away from the origin (e.g. by replacing*x*2by (x_{−}*x*0)2in its equation). It can
be proved11that*every*quadric surface can be translated and/or rotated so that its equation
matches one of the six types that we described. For example, *z*_{=}2x yis a case of equation
(1.33) with “mixed” variables, e.g. with*D*_{6=}0 so that we get an*x y*term. This equation does
not match any of the types we considered. However, by rotating the*x- and* *y-axes by 45*◦in
the*x y-plane by means of the coordinate transformationx*_{=}(x′_{−}*y*′)/p2,*y*_{=}(x′_{+}*y*′)/p2,*z*_{=}*z*′,
then *z*_{=}2x ybecomes the hyperbolic paraboloid *z*′_{=}(x′)2_{−}(y′)2 in the (x′,*y*′,*z*′) coordinate
system. That is,*z*_{=}2x yis a hyperbolic paraboloid as in equation (1.38), but rotated 45◦in
the*x y-plane.*

**Exercises**

**A**

For Exercises 1-4, determine if the given equation describes a sphere. If so, find its radius and center.

**1.** *x*2_{+}*y*2_{+}*z*2_{−}4x_{−}6*y*_{−}10z_{+}37_{=}0 **2.** *x*2_{+}*y*2_{+}*z*2_{+}2x_{−}2*y*_{−}8z_{+}19_{=}0

**3.** 2x2_{+}2*y*2_{+}2*z*2_{+}4x_{+}4*y*_{+}4*z*_{−}44_{=}0 **4.** *x*2_{+}*y*2_{−}*z*2_{+}12x_{+}2*y*_{−}4z_{+}32_{=}0

**5.** Find the point(s) of intersection of the sphere (x_{−}3)2_{+}(*y*_{+}1)2_{+}(z_{−}3)2_{=}9 and the line
*x*_{= −}1_{+}2t, *y*_{= −}2_{−}3t, *z*_{=}3_{+}*t.*

**B**

**6.** Find the intersection of the spheres*x*2_{+}*y*2_{+}*z*2_{=}9 and (x_{−}4)2_{+}(y_{+}2)2_{+}(z_{−}4)2_{=}9.

**7.** Find the intersection of the sphere*x*2_{+}*y*2_{+}*z*2_{=}9 and the cylinder *x*2_{+}*y*2_{=}4.

**8.** Find the trace of the hyperboloid of one sheet * _{a}x*22+

*y*2

*b*2−

*z*2

*c*2 =1 in the plane*x*=*a, and the*
trace in the plane *y*_{=}*b.*

**9.** Find the trace of the hyperbolic paraboloid *x*2

*a*2−

*y*2

*b*2=*z _{c}* in the

*x y-plane.*

**C**

**10.** It can be shown that any four*noncoplanar*points (i.e. points that do not lie in the same
plane) determine a sphere.12 Find the equation of the sphere that passes through the
points (0, 0, 0), (0, 0, 2), (1,_{−}4, 3) and (0,_{−}1, 3).*(Hint: Equation (1.31))*

**11.** Show that the hyperboloid of one sheet is a doubly ruled surface, i.e. each point on
the surface is on two lines lying entirely on the surface. *(Hint: Write equation (1.35) as*

*x*2

*a*2−

*z*2

*c*2=1−

*y*2

*b*2 *, factor each side. Recall that two planes intersect in a line.)*

**12.** Show that the hyperbolic paraboloid is a doubly ruled surface.*(Hint: Exercise 11)*

*y*
*z*
*x*
0
(0, 0, 2)
(*x*,*y*, 0)
(*a*,*b*,*c*)
1 *S*
**Figure 1.6.10**

**13.** Let *S* be the sphere with radius 1 centered at (0, 0, 1),
and let*S*∗be*S*without the “north pole” point (0, 0, 2). Let
(a,*b,c) be an arbitrary point on* *S*∗. Then the line passing
through (0, 0, 2) and (a,*b,c) intersects thex y-plane at some*
point (x,*y, 0), as in Figure 1.6.10. Find this point (x,y, 0) in*
terms of*a,b*and*c.*

(Note: Every point in the *x y-plane can be matched with a*
point on*S*∗, and vice versa, in this manner. This method is
called*stereographic projection, which essentially identifies*
all of_{R}2 with a “punctured” sphere.)