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Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation z = y 2 .


This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.

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When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in [link] .

Definition

The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.

This figure has two images. The first image is a surface. A cross section of the surface parallel to the x z plane would be a sine curve. The second image is the sine curve in the x y plane.
(a) This is one view of the graph of equation z = sin x . (b) To find the trace of the graph in the xz -plane, set y = 0 . The trace is simply a two-dimensional sine wave.

Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in [link] , that the trace of the graph of z = sin x in the xz -plane is useful in constructing the graph. The trace in the xy -plane, though, is just a series of parallel lines, and the trace in the yz -plane is simply one line.

Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.

Quadric surfaces

We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.

Definition

Quadric surfaces are the graphs of equations that can be expressed in the form

A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + J z + K = 0 .

When a quadric surface intersects a coordinate plane, the trace is a conic section.

An ellipsoid    is a surface described by an equation of the form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . Set x = 0 to see the trace of the ellipsoid in the yz -plane. To see the traces in the y - and xz -planes, set z = 0 and y = 0 , respectively. Notice that, if a = b , the trace in the xy -plane is a circle. Similarly, if a = c , the trace in the xz -plane is a circle and, if b = c , then the trace in the yz -plane is a circle. A sphere, then, is an ellipsoid with a = b = c .

Sketching an ellipsoid

Sketch the ellipsoid x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 .

Start by sketching the traces. To find the trace in the xy -plane, set z = 0 : x 2 2 2 + y 2 3 2 = 1 (see [link] ). To find the other traces, first set y = 0 and then set x = 0 .

This figure has three images. The first image is an oval centered around the origin of the rectangular coordinate system. It intersects the x axis at -2 and 2. It intersects the y-axis at -3 and 3. The second image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -2 and 2 and the y-axis at -5 and 5. The third image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -3 and 3 and the y-axis at -5 and 5.
(a) This graph represents the trace of equation x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 in the xy -plane, when we set z = 0 . (b) When we set y = 0 , we get the trace of the ellipsoid in the xz -plane, which is an ellipse. (c) When we set x = 0 , we get the trace of the ellipsoid in the yz -plane, which is also an ellipse.

Now that we know what traces of this solid look like, we can sketch the surface in three dimensions ( [link] ).

This figure has two images. The first image is a vertical ellipse. There two curves drawn with dashed lines around the center horizontally and vertically to give the image a 3-dimensional shape. The second image is a solid elliptical shape with the center at the origin of the 3-dimensional coordinate system.
(a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.
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The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form x 2 a 2 + y 2 b 2 = z c , then we call that surface an elliptic paraboloid    . The trace in the xy -plane is an ellipse, but the traces in the xz -plane and yz -plane are parabolas ( [link] ). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation x 2 a 2 + z 2 c 2 = y b or y 2 b 2 + z 2 c 2 = x a .

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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