2 We have already looked at two special types of surfaces: VECTORS AND THE GEOMETRY OF SPACEWe have already looked at two special types of surfaces:Planes (Section 12.5)Spheres (Section 12.1)
3 Here, we investigate two other types of surfaces: VECTORS AND THE GEOMETRY OF SPACEHere, we investigate two other types of surfaces:CylindersQuadric surfaces
4 Cylinders and Quadric Surfaces VECTORS AND THE GEOMETRY OF SPACE12.6Cylinders and Quadric SurfacesIn this section, we will learn about:Cylinders and various types of quadric surfaces.
5 TRACESTo sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes.These curves are called traces (or cross-sections) of the surface.
6 CYLINDERA cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve.
7 Sketch the graph of the surface z = x2 CYLINDERSExample 1Sketch the graph of the surface z = x2Notice that the equation of the graph, z = x2, doesn’t involve y.This means that any vertical plane with equation y = k (parallel to the xz-plane) intersects the graph in a curve with equation z = x2.So, these vertical traces are parabolas.
8 CYLINDERSExample 1The figure shows how the graph is formed by taking the parabola z = x2 in the xz-plane and moving it in the direction of the y-axis.
9 PARABOLIC CYLINDERExample 1The graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola.Here, the rulings of the cylinder are parallel to the y-axis.
10 CYLINDERSIn Example 1, we noticed the variable y is missing from the equation of the cylinder.This is typical of a surface whose rulings are parallel to one of the coordinate axes.If one of the variables x, y, or z is missing from the equation of a surface, then the surface is a cylinder.
11 Identify and sketch the surfaces. CYLINDERSExample 2Identify and sketch the surfaces.x2 + y2 = 1y2 + z2 = 1
12 CYLINDERSExample 2 aHere, z is missing and the equations x2 + y2 = 1, z = k represent a circle with radius 1 in the plane z = k.
13 CYLINDERSExample 2 aThus, the surface x2 + y2 = 1 is a circular cylinder whose axis is the z-axis.Here, the rulings are vertical lines.
14 CYLINDERSExample 2 bIn this case, x is missing and the surface is a circular cylinder whose axis is the x-axis.It is obtained by taking the circle y2 + z2 = 1, x = 0 in the yz-plane, and moving it parallel to the x-axis.
15 CYLINDERSNoteWhen you are dealing with surfaces, it is important to recognize that an equation like x2 +y2 = 1 represents a cylinder and not a circle.The trace of the cylinder x2 + y2 = 1 in the xy-plane is the circle with equations x2 + y2 = 1, z = 0
16 QUADRIC SURFACEA quadric surface is the graph of a second-degree equation in three variables x, y, and z.
17 The most general such equation is: QUADRIC SURFACESThe most general such equation is:Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0A, B, C, …, J are constants.
18 QUADRIC SURFACESHowever, by translation and rotation, it can be brought into one of the two standard forms:Ax2 + By2 + Cz2 + J = 0Ax2 + By2 + Iz = 0
19 QUADRIC SURFACESQuadric surfaces are the counterparts in three dimensions of the conic sections in the plane.See Section 10.5 for a review of conic sections.
20 Use traces to sketch the quadric surface with equation QUADRIC SURFACESExample 3Use traces to sketch the quadric surface with equation
21 By substituting z = 0, we find that the trace in the xy-plane is: QUADRIC SURFACESExample 3By substituting z = 0, we find that the trace in the xy-plane is:x2 + y2/9 = 1We recognize this as an equation of an ellipse.
22 In general, the horizontal trace in the plane z = k is: QUADRIC SURFACESExample 3In general, the horizontal trace in the plane z = k is:This is an ellipse—provided that k2 < 4, that is, –2 < k < 2.
23 Similarly, the vertical traces are also ellipses: QUADRIC SURFACESExample 3Similarly, the vertical traces are also ellipses:
24 QUADRIC SURFACESExample 3The figure shows how drawing some traces indicates the shape of the surface.
25 It’s called an ellipsoid because all of its traces are ellipses. Example 3It’s called an ellipsoid because all of its traces are ellipses.
26 Notice that it is symmetric with respect to each coordinate plane. QUADRIC SURFACESExample 3Notice that it is symmetric with respect to each coordinate plane.This is a reflection of the fact that its equation involves only even powers of x, y, and z.
27 Use traces to sketch the surface z = 4x2 + y2 QUADRIC SURFACESExample 4Use traces to sketch the surface z = 4x2 + y2
28 QUADRIC SURFACESExample 4If we put x = 0, we get z = y2So, the yz-plane intersects the surface in a parabola.
29 If we put x = k (a constant), we get z = y2 + 4k2 QUADRIC SURFACESExample 4If we put x = k (a constant), we get z = y2 + 4k2This means that, if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward.
30 Similarly, if y = k, the trace is z = 4x2 + k2 QUADRIC SURFACESExample 4Similarly, if y = k, the trace is z = 4x2 + k2This is again a parabola that opens upward.
31 If we put z = k, we get the horizontal traces 4x2 + y2 = k QUADRIC SURFACESExample 4If we put z = k, we get the horizontal traces 4x2 + y2 = kWe recognize this as a family of ellipses.
32 Knowing the shapes of the traces, we can sketch the graph as below. QUADRIC SURFACESExample 4Knowing the shapes of the traces, we can sketch the graph as below.
33 ELLIPTIC PARABOLOIDExample 4Due to the elliptical and parabolic traces, the quadric surface z = 4x2 +y2 is called an elliptic paraboloid.Horizontal traces are ellipses.Vertical traces are parabolas.
34 Sketch the surface z = y2 – x2 QUADRIC SURFACESExample 5Sketch the surface z = y2 – x2
35 QUADRIC SURFACESExample 5The traces in the vertical planes x = k are the parabolas z = y2 – k2, which open upward.
36 QUADRIC SURFACESExample 5The traces in y = k are the parabolas z = –x2 + k2, which open downward.
37 The horizontal traces are y2 – x2 = k, a family of hyperbolas. QUADRIC SURFACESExample 5The horizontal traces are y2 – x2 = k,a family of hyperbolas.
38 All traces are labeled with the value of k. QUADRIC SURFACESExample 5All traces are labeled with the value of k.
39 QUADRIC SURFACESExample 5Here, we show how the traces appear when placed in their correct planes.
40 HYPERBOLIC PARABOLOID Example 5Here, we fit together the traces from the previous figure to form the surface z = y2 – x2, a hyperbolic paraboloid.
41 HYPERBOLIC PARABOLOID Example 5Notice that the shape of the surface near the origin resembles that of a saddle.This surface will be investigated further in Section 14.7 when we discuss saddle points.
43 The trace in any horizontal plane z = k is the ellipse QUADRIC SURFACESExample 6The trace in any horizontal plane z = k is the ellipse
44 The traces in the xz- and yz-planes are the hyperbolas QUADRIC SURFACESExample 6The traces in the xz- and yz-planes are the hyperbolas
45 HYPERBOLOID OF ONE SHEET Example 6This surface is called a hyperboloid of one sheet.
46 GRAPHING SOFTWAREThe idea of using traces to draw a surface is employed in three-dimensional (3-D) graphing software for computers.
47 GRAPHING SOFTWAREIn most such software,Traces in the vertical planes x = k and y = k are drawn for equally spaced values of k.Parts of the graph are eliminated using hidden line removal.
48 GRAPHING SOFTWARENext, we show computer-drawn graphs of the six basic types of quadric surfaces in standard form.All surfaces are symmetric with respect to the z-axis.If a surface is symmetric about a different axis, its equation changes accordingly.
55 We collect the graphs in this table. GRAPHING SOFTWAREWe collect the graphs in this table.
56 Identify and sketch the surface 4x2 – y2 + 2z2 +4 = 0 QUADRIC SURFACESExample 7Identify and sketch the surface4x2 – y2 + 2z2 +4 = 0
57 Dividing by –4, we first put the equation in standard form: QUADRIC SURFACESExample 7Dividing by –4, we first put the equation in standard form:
58 QUADRIC SURFACESExample 7Comparing the equation with the table, we see that it represents a hyperboloid of two sheets.The only difference is that, in this case, the axis of the hyperboloid is the y-axis.
59 The traces in the xy- and yz-planes are the hyperbolas QUADRIC SURFACESExample 7The traces in the xy- and yz-planes are the hyperbolas
60 The surface has no trace in the xz-plane. QUADRIC SURFACESExample 7The surface has no trace in the xz-plane.
61 QUADRIC SURFACESExample 7However, traces in the vertical planes y = k for |k| > 2 are the ellipsesThis can be written as:
62 Those traces are used to make this sketch. QUADRIC SURFACESExample 7Those traces are used to make this sketch.
63 Classify the quadric surface QUADRIC SURFACESExample 8Classify the quadric surfacex2 + 2z2 – 6x – y + 10 = 0
64 By completing the square, we rewrite the equation as: QUADRIC SURFACESExample 8By completing the square, we rewrite the equation as:y – 1 = (x – 3)2 + 2z2
65 QUADRIC SURFACESExample 8Comparing the equation with the table, we see that it represents an elliptic paraboloid.However, the axis of the paraboloid is parallel to the y-axis, and it has been shifted so that its vertex is the point (3, 1, 0).
66 QUADRIC SURFACESExample 8The traces in the plane y = k (k > 1) are the ellipses (x – 3)2 + 2z2 = k – y = k
67 QUADRIC SURFACESExample 8The trace in the xy-plane is the parabola with equation y = 1 + (x – 3)2, z = 0
68 The paraboloid is sketched here. QUADRIC SURFACESExample 8The paraboloid is sketched here.
69 APPLICATIONS OF QUADRIC SURFACES Examples of quadric surfaces can be found in the world around us.In fact, the world itself is a good example.
70 APPLICATIONS OF QUADRIC SURFACES Though the earth is commonly modeled as a sphere, a more accurate model is an ellipsoid.This is because the earth’s rotation has caused a flattening at the poles.See Exercise 47.
71 APPLICATIONS OF QUADRIC SURFACES Circular paraboloids—obtained by rotating a parabola about its axis—are used to collect and reflect light, sound, and radio and television signals.
72 APPLICATIONS OF QUADRIC SURFACES For instance, in a radio telescope, signals from distant stars that strike the bowl are reflected to the receiver at the focus and are therefore amplified.The idea is explained in Problem 18 in Chapter 3
73 APPLICATIONS OF QUADRIC SURFACES The same principle applies to microphones and satellite dishes in the shape of paraboloids.
74 APPLICATIONS OF QUADRIC SURFACES Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of one sheet for reasons of structural stability.
75 APPLICATIONS OF QUADRIC SURFACES Pairs of hyperboloids are used to transmit rotational motion between skew axes.
76 APPLICATIONS OF QUADRIC SURFACES Finally, the cogs of gears are the generating lines of the hyperboloids.