11.6 Cylinders and Quadric Surfaces

Arkansas Tech University MATH 2934: Calculus III Dr. Marcel B Finan

11.6 Space Surfaces: Cylinders and Quadric Surfaces By a space surface we mean the graph of an equation in the variables x, y, z : f(x, y, z) = 0. We have already encountered two space surfaces: spheres and planes. Recall the equation of a sphere centered at (a, b, c) and radius R is

(x − a)2 + (y − b)2 + (z − c)2 = R2.

This is an equation of the form f(x, y, z) = 0 where f(x, y, z) = (x−a)2 +(y− b)2+(z−c)2−R2. Likewise, the general equation of a plane ax+by+cz+d = 0 can be written in the form f(x, y, z) = 0 where f(x, y, z) = ax + by + cz + d. Next, the intersection of a plane parallel to a coordinate plane with a surface is called a trace or a cross-section. For example, the trace of the sphere x2 + y2 + z2 = 100 with a plane z = 1 parallel to the xy−plane is the circle x2 + y2 = 99. By a cylindrical surface or simply a cylinder we mean a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a plane curve (called the generator). See Figure 11.6.1.

Figure 11.6.1 A right cylinder is a cylinder where the generator is in one of the coordinate planes and the rulings are perpendicular to the coordinate plane containing the generator. Thus, the rulings are parallel to a coordinate axis. The variable associated with the coordinate axis does not appear in the equation of the cylinder. The following example describes three types of right cylinders.

Example 11.6.1 x2 y2 Sketch the graph of each of the following cylindrical surfaces: (a) a2 + b2 = 1 2 x2 y2 (b) x = ay , a > 0 (c) a2 − b2 = 1. Solution. (a) This surface is called an elliptic cylinder. The rulings are parallel to

1 the missing variable which is in this case the z−axis. The generator is an ellipse in the xy− plane. See Figure 11.6.2(a). Vertical traces are union of two parallel lines that are parallel to the missing coordinate axis whereas horizontal traces are ellipses. (b) This surface is called a parabolic cylinder. The rulings are parallel to the missing variable which is in this case the z−axis. The generator is a parabola in the xy− plane. See Figure 11.6.2(b). Horizontal traces are parabolas. Vertical traces are the union of two parallel lines. Notice that the axis of symmetry is the variable with the ﬁrst power. (c) This surface is called a hyperbolic cylinder. The rulings are parallel to the missing variable which is in this case the z−axis. The generator is a hyperbola. See Figure 11.6.2(c). Horizontal traces are hyperbolas. The vertex of the generator is on the axis with a plus sign which is in this case the x−axis

Figure 11.6.2

Quadric Surfaces Quadric surfaces are the counterparts in three dimensions of the conic sec- tions in the plane. A quadric surface is the graph of a second-degree equation in the variables x, y, z given by

Ax2 + By2 + Cz2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0.

There is no way that we can possibly list all of them, but there are some standard equations so here is a list of some of the more common quadric

2 surfaces. x2 y2 z2 An ellipsoid is the surface with equation a2 + b2 + c2 = 1. See Figure 11.6.3. Intersections with any plane parallel to a coordinate plane (i.e., traces) give an ellipse.

Figure 11.6.3

Notice that a sphere centered at the origin and with radius a is a special case of an ellipsoid where a = b = c. z x2 y2 An elliptic paraboloid is the surface with equation c = a2 + b2 . See Figure 11.6.4. Vertical traces are parabolas whereas horizontal traces are ellipses. The axis of the paraboloid corresponds to the variable raised to the ﬁrst power.

Figure 11.6.4

z x2 y2 A hyperbolic paraboloid is the surface with equation c = a2 − b2 . See Figure 11.6.5. vertical traces are parabolas whereas horizontal traces are hyperbolas. Notice that the shape of the surface near the origin resembles that of a saddle. Notice that the variable raised to the ﬁrst power indicates the axis of symmetry.

3 Figure 11.6.5

x2 y2 z2 A hyperboloid of one sheet is the surface with equation a2 + b2 − c2 = 1. See Figure 11.6.6. Vertical traces are hyperbolas whereas horizontal traces are ellipses. The axis of symmetry corresponds to the variable whose coeﬃ- cient is negative.

Figure 11.6.6

x2 y2 z2 A hyperboloid of two sheets is the surface with equation a2 − b2 − c2 = 1. See Figure 11.6.7. Vertical traces are ellipses whereas horizontal traces are hyperbolas. The two minus signs indicate two sheets and the axis of symmetry is the variable with a plus sign.

4 Figure 11.6.7

z2 A cone with axis of symmetry the z−axis is the surface with equation c2 = x2 y2 a2 + b2 . See Figure 11.6.8. Vertical traces are hyperbolas whereas horizontal traces are ellipses.

Figure 11.6.8

Example 11.6.2 Classify and sketch the quadric surface x2 + 2z2 − 6x − y + 10 = 0.

Solution. By completing the square, we have (x − 3)2 + 2z2 − y + 1 = 0 or y − 1 = (x − 3)2 + 2z2. This is an elliptic paraboloid with axis parallel to the y−axis and it has been shifted so that its vertex is the point (3, 1, 0). See Figure 11.6.9

5 Figure 11.6.9

Example 11.6.3 Classify and sketch the quadric surface 4x2 − 3y2 + 12z2 + 12 = 0.

Solution. Writing the equation in standard form, we ﬁnd

y2 x2 − − z2 = 1. 4 3 The surface is a hyperboloid with two sheets with axis of symmetry the y−axis as shown in Figure 11.6.10.

Figure 11.6.10

6 Surfaces of Revolution When a plane curve is revolved about a line, it generates a surface called a surface of revolution. For example, if the curve y = r(z) is rotated about the z−axis then it generates the surface of revolution shown in Figure 11.6.11.

Figure 11.6.11 The cross-section of the surface in the plane at height z is a circle whose equation is x2 + y2 = [r(z)]2 which is the equation of the surface of revolution since z is arbitrary. Likewise, if the generator curve is rotated avout the x− axis then the equation of the surface of revolution is y2 + z2 = [r(x)]2. If the generator is rotated about the −axis then the equation of the surface of revolution is x2 + z2 = [r(y)]2.

Example 11.6.4 Find an equation for the surface of revolution formed by revolving the curve 9x2 = y3 about the y−axis.

Solution. The equation of the surface of revolution (see Figure 11.6.12) is 2 2 2 1 3 1 3 x + z = y 2 = y 3 9

7 Figure 11.6.12 Remark 11.6.1 If the equation of the surface of revolution is given, say for example x2 +z2 = [r(y)]2. Then the axis of rotation is the y−axis. However, there are two possible generating curves, namely, x = r(y) or z = r(y). Note that x and z are the variables on the left-side of the equation. In the previous example, we were given the generating curve and the line of rotation and we were asked to ﬁnd the equation of the surface of revolution. In the next example, we go the other way. Given the equation of the surface we want to ﬁnd a generating curve and the line of rotation. Example 11.6.5 Find a generating curve and the axis of revolution for the surface: x2 + 3y2 + z2 = 9. Solution. The given equation can be written in the form x2 y2 z2 + + = 1 9 3 9 8 which is an ellipsoid as shown in Figure 11.6.13.

Figure 11.6.13

Since x2 and z2 have the same coeﬃcient, the equation of the surface of revolution is x2 + z2 = 9 − 3y2. Thus, the axis of revolution is the y−axis. A generating curve can be either z = p9 − 3y2 or x = p9 − 3y2