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12 VECTORS AND THE GEOMETRY OF SPACE

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**We have already looked at two special types of surfaces:**

VECTORS AND THE GEOMETRY OF SPACE We have already looked at two special types of surfaces: Planes (Section 12.5) Spheres (Section 12.1)

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**Here, we investigate two other types of surfaces:**

VECTORS AND THE GEOMETRY OF SPACE Here, we investigate two other types of surfaces: Cylinders Quadric surfaces

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**Cylinders and Quadric Surfaces**

VECTORS AND THE GEOMETRY OF SPACE 12.6 Cylinders and Quadric Surfaces In this section, we will learn about: Cylinders and various types of quadric surfaces.

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TRACES To 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.

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CYLINDER A 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.

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**Sketch the graph of the surface z = x2**

CYLINDERS Example 1 Sketch the graph of the surface z = x2 Notice 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.

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CYLINDERS Example 1 The 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.

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PARABOLIC CYLINDER Example 1 The 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.

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CYLINDERS In 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.

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**Identify and sketch the surfaces.**

CYLINDERS Example 2 Identify and sketch the surfaces. x2 + y2 = 1 y2 + z2 = 1

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CYLINDERS Example 2 a Here, z is missing and the equations x2 + y2 = 1, z = k represent a circle with radius 1 in the plane z = k.

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CYLINDERS Example 2 a Thus, the surface x2 + y2 = 1 is a circular cylinder whose axis is the z-axis. Here, the rulings are vertical lines.

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CYLINDERS Example 2 b In 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.

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CYLINDERS Note When 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

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QUADRIC SURFACE A quadric surface is the graph of a second-degree equation in three variables x, y, and z.

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**The most general such equation is: **

QUADRIC SURFACES The most general such equation is: Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0 A, B, C, …, J are constants.

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QUADRIC SURFACES However, by translation and rotation, it can be brought into one of the two standard forms: Ax2 + By2 + Cz2 + J = 0 Ax2 + By2 + Iz = 0

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QUADRIC SURFACES Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane. See Section 10.5 for a review of conic sections.

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**Use traces to sketch the quadric surface with equation**

QUADRIC SURFACES Example 3 Use traces to sketch the quadric surface with equation

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**By substituting z = 0, we find that the trace in the xy-plane is:**

QUADRIC SURFACES Example 3 By substituting z = 0, we find that the trace in the xy-plane is: x2 + y2/9 = 1 We recognize this as an equation of an ellipse.

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**In general, the horizontal trace in the plane z = k is:**

QUADRIC SURFACES Example 3 In general, the horizontal trace in the plane z = k is: This is an ellipse—provided that k2 < 4, that is, –2 < k < 2.

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**Similarly, the vertical traces are also ellipses:**

QUADRIC SURFACES Example 3 Similarly, the vertical traces are also ellipses:

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QUADRIC SURFACES Example 3 The figure shows how drawing some traces indicates the shape of the surface.

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**It’s called an ellipsoid because all of its traces are ellipses.**

Example 3 It’s called an ellipsoid because all of its traces are ellipses.

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**Notice that it is symmetric with respect to each coordinate plane.**

QUADRIC SURFACES Example 3 Notice 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.

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**Use traces to sketch the surface z = 4x2 + y2**

QUADRIC SURFACES Example 4 Use traces to sketch the surface z = 4x2 + y2

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QUADRIC SURFACES Example 4 If we put x = 0, we get z = y2 So, the yz-plane intersects the surface in a parabola.

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**If we put x = k (a constant), we get z = y2 + 4k2**

QUADRIC SURFACES Example 4 If we put x = k (a constant), we get z = y2 + 4k2 This means that, if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward.

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**Similarly, if y = k, the trace is z = 4x2 + k2**

QUADRIC SURFACES Example 4 Similarly, if y = k, the trace is z = 4x2 + k2 This is again a parabola that opens upward.

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**If we put z = k, we get the horizontal traces 4x2 + y2 = k**

QUADRIC SURFACES Example 4 If we put z = k, we get the horizontal traces 4x2 + y2 = k We recognize this as a family of ellipses.

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**Knowing the shapes of the traces, we can sketch the graph as below.**

QUADRIC SURFACES Example 4 Knowing the shapes of the traces, we can sketch the graph as below.

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ELLIPTIC PARABOLOID Example 4 Due 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.

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**Sketch the surface z = y2 – x2**

QUADRIC SURFACES Example 5 Sketch the surface z = y2 – x2

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QUADRIC SURFACES Example 5 The traces in the vertical planes x = k are the parabolas z = y2 – k2, which open upward.

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QUADRIC SURFACES Example 5 The traces in y = k are the parabolas z = –x2 + k2, which open downward.

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**The horizontal traces are y2 – x2 = k, a family of hyperbolas.**

QUADRIC SURFACES Example 5 The horizontal traces are y2 – x2 = k, a family of hyperbolas.

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**All traces are labeled with the value of k.**

QUADRIC SURFACES Example 5 All traces are labeled with the value of k.

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QUADRIC SURFACES Example 5 Here, we show how the traces appear when placed in their correct planes.

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**HYPERBOLIC PARABOLOID**

Example 5 Here, we fit together the traces from the previous figure to form the surface z = y2 – x2, a hyperbolic paraboloid.

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**HYPERBOLIC PARABOLOID**

Example 5 Notice 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.

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QUADRIC SURFACES Example 6 Sketch the surface

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**The trace in any horizontal plane z = k is the ellipse**

QUADRIC SURFACES Example 6 The trace in any horizontal plane z = k is the ellipse

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**The traces in the xz- and yz-planes are the hyperbolas**

QUADRIC SURFACES Example 6 The traces in the xz- and yz-planes are the hyperbolas

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**HYPERBOLOID OF ONE SHEET**

Example 6 This surface is called a hyperboloid of one sheet.

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GRAPHING SOFTWARE The idea of using traces to draw a surface is employed in three-dimensional (3-D) graphing software for computers.

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GRAPHING SOFTWARE In 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.

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GRAPHING SOFTWARE Next, 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.

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ELLIPSOID

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CONE

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ELLIPTIC PARABOLOID

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**HYPERBOLOID OF ONE SHEET**

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**HYPERBOLIC PARABOLOID**

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**HYPERBOLOID OF TWO SHEETS**

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**We collect the graphs in this table.**

GRAPHING SOFTWARE We collect the graphs in this table.

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**Identify and sketch the surface 4x2 – y2 + 2z2 +4 = 0**

QUADRIC SURFACES Example 7 Identify and sketch the surface 4x2 – y2 + 2z2 +4 = 0

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**Dividing by –4, we first put the equation in standard form:**

QUADRIC SURFACES Example 7 Dividing by –4, we first put the equation in standard form:

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QUADRIC SURFACES Example 7 Comparing 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.

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**The traces in the xy- and yz-planes are the hyperbolas**

QUADRIC SURFACES Example 7 The traces in the xy- and yz-planes are the hyperbolas

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**The surface has no trace in the xz-plane.**

QUADRIC SURFACES Example 7 The surface has no trace in the xz-plane.

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QUADRIC SURFACES Example 7 However, traces in the vertical planes y = k for |k| > 2 are the ellipses This can be written as:

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**Those traces are used to make this sketch.**

QUADRIC SURFACES Example 7 Those traces are used to make this sketch.

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**Classify the quadric surface**

QUADRIC SURFACES Example 8 Classify the quadric surface x2 + 2z2 – 6x – y + 10 = 0

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**By completing the square, we rewrite the equation as:**

QUADRIC SURFACES Example 8 By completing the square, we rewrite the equation as: y – 1 = (x – 3)2 + 2z2

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QUADRIC SURFACES Example 8 Comparing 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).

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QUADRIC SURFACES Example 8 The traces in the plane y = k (k > 1) are the ellipses (x – 3)2 + 2z2 = k – y = k

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QUADRIC SURFACES Example 8 The trace in the xy-plane is the parabola with equation y = 1 + (x – 3)2, z = 0

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**The paraboloid is sketched here.**

QUADRIC SURFACES Example 8 The paraboloid is sketched here.

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**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.

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**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.

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**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.

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**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

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**APPLICATIONS OF QUADRIC SURFACES**

The same principle applies to microphones and satellite dishes in the shape of paraboloids.

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**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.

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**APPLICATIONS OF QUADRIC SURFACES**

Pairs of hyperboloids are used to transmit rotational motion between skew axes.

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**APPLICATIONS OF QUADRIC SURFACES**

Finally, the cogs of gears are the generating lines of the hyperboloids.

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