drawing cones from their equations in 3d

12.6: Quadric Surfaces

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• Professor (Mathematics) at Massachusetts Institute of Technology (Strang) & University of Wisconsin-Stevens Point (Herman)
• Source: OpenStax CNX

Learning Objectives

• Place a cylinder equally a type of three-dimensional surface.
• Recognize the chief features of ellipsoids, paraboloids, and hyperboloids.
• Use traces to draw the intersections of quadric surfaces with the coordinate planes.

We have been exploring vectors and vector operations in three-dimensional space, and we accept developed equations to describe lines, planes, and spheres. In this section, we use our noesis of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a diverseness of other surfaces that tin can exist graphed in a 3-dimensional coordinate organisation.

Identifying Cylinders

The offset surface we'll examine is the cylinder. Although almost people immediately think of a hollow piping or a soda harbinger when they hear the discussion cylinder, hither nosotros utilise the broad mathematical pregnant of the term. As we have seen, cylindrical surfaces don't have to be circular. A rectangular heating duct is a cylinder, as is a rolled-upward yoga mat, the cross-section of which is a spiral shape.

In the two-dimensional coordinate plane, the equation \( x^2+y^2=9\) describes a circle centered at the origin with radius \( three\). In iii-dimensional space, this aforementioned equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the \(z\)-axis (Effigy \(\PageIndex{i}\)), forming a hollow tube. We can then construct a cylinder from the ready of lines parallel to the \(z\)-centrality passing through the circle \( ten^2+y^2=9\) in the \(xy\)-plane, as shown in the figure. In this way, whatsoever curve in one of the coordinate planes can be extended to become a surface.

Effigy \(\PageIndex{1}\): In three-dimensional infinite, the graph of equation \( x^2+y^2=9\) is a cylinder with radius \( 3\) centered on the \(z\)-centrality. It continues indefinitely in the positive and negative directions.

Definition: cylinders and rulings

A fix of lines parallel to a given line passing through a given bend is known as a cylindrical surface, or cylinder . The parallel lines are called rulings.

From this definition, nosotros tin can come across that we nonetheless have a cylinder in three-dimensional space, even if the curve is not a circle. Any bend can grade a cylinder, and the rulings that compose the cylinder may be parallel to whatsoever given line (Figure \(\PageIndex{2}\)).

Figure \(\PageIndex{2}\): In three-dimensional space, the graph of equation \( z=ten^3\) is a cylinder, or a cylindrical surface with rulings parallel to the \(y\)-axis.

Case \( \PageIndex{one}\): Graphing Cylindrical Surfaces

Sketch the graphs of the following cylindrical surfaces.

1. \( x^2+z^two=25\)
2. \( z=2x^2−y\)
3. \( y=\sin x\)

Solution

a. The variable \( y\) can take on whatsoever value without limit. Therefore, the lines ruling this surface are parallel to the \(y\)-axis. The intersection of this surface with the \(xz\)-plane forms a circumvolve centered at the origin with radius \( 5\) (run into Effigy \(\PageIndex{3}\)).

Effigy \(\PageIndex{3}\): The graph of equation \( x^2+z^ii=25\) is a cylinder with radius \( v\) centered on the \(y\)-axis.

b. In this example, the equation contains all three variables —\( 10,y,\) and \( z\)— so none of the variables can vary arbitrarily. The easiest fashion to visualize this surface is to utilise a estimator graphing utility (Effigy \(\PageIndex{4}\)).

Figure \(\PageIndex{4}\)

c. In this equation, the variable \( z\) tin can take on whatsoever value without limit. Therefore, the lines composing this surface are parallel to the \(z\)-axis. The intersection of this surface with the xy-aeroplane outlines curve \( y=\sin x\) (Figure \(\PageIndex{5}\)).

Figure \(\PageIndex{5}\): The graph of equation \( y=\sin x\) is formed by a set of lines parallel to the \(z\)-centrality passing through curve \( y=\sin x\) in the \(xy\)-aeroplane.

Exercise \( \PageIndex{ane}\):

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

Hint

The variable \( x\) can take on any value without limit.

Answer

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. Nosotros can come across them in the plot of the cylinder in Figure \(\PageIndex{6}\).

Definition: traces

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

Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, just i set of traces is useful. Notice, in Figure \(\PageIndex{6}\), that the trace of the graph of \( z=\sin ten\) 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 1 line.

Figure \(\PageIndex{6}\): (a) This is one view of the graph of equation \( z=\sin x\). (b) To detect the trace of the graph in the \(xz\)-airplane, set \( y=0\). The trace is only a ii-dimensional sine wave.

Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed and so simply, all the same. Nosotros now explore more circuitous surfaces, and traces are an of import tool in this investigation.

Quadric Surfaces

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

Definition: Quadric surfaces and conic sections

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

\[Ax^ii+By^ii+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.\]

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 \( \dfrac{x^ii}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^ii}{c^two}=ane.\) Set up \( ten=0\) to run across the trace of the ellipsoid in the yz-aeroplane. To see the traces in the \(xy\)- and \(xz\)-planes, ready \( z=0\) 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 \(xz\)-airplane is a circle and, if \( b=c\), then the trace in the \(yz\)-plane is a circle. A sphere, then, is an ellipsoid with \( a=b=c.\)

Example \( \PageIndex{2}\): Sketching an Ellipsoid

Sketch the ellipsoid

\[ \dfrac{x^2}{two^2}+\dfrac{y^2}{3^2}+\dfrac{z^ii}{v^2}=ane.\]

Solution

Start by sketching the traces. To find the trace in the xy-plane, ready \( z=0: \dfrac{ten^two}{2^2}+\dfrac{y^2}{3^2}=ane\) (Figure \(\PageIndex{vii}\)). To observe the other traces, first set \( y=0\) and and so set \( x=0.\)

Figure \(\PageIndex{7}\): (a) This graph represents the trace of equation \( \dfrac{x^2}{2^2}+\dfrac{y^two}{three^2}+\dfrac{z^2}{5^ii}=1\) in the \(xy\)-plane, when we gear up \( z=0\). (b) When we prepare \( y=0\), we get the trace of the ellipsoid in the \(xz\)-plane, which is an ellipse. (c) When we fix \( x=0\), we get the trace of the ellipsoid in the \(yz\)-plane, which is besides an ellipse.

At present that we know what traces of this solid look like, we tin sketch the surface in three dimensions (Effigy \(\PageIndex{eight}\)).

Figure \(\PageIndex{8}\): (a) The traces provide a framework for the surface. (b) The eye of this ellipsoid is the origin.

The trace of an ellipsoid is an ellipse in each of the coordinate planes. Still, this does not take to exist the example for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated past the proper name of the surface. For example, if a surface can be described by an equation of the grade

\[ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\dfrac{z}{c}\]

then nosotros call that surface an elliptic paraboloid. The trace in the xy-plane is an ellipse, only the traces in the xz-aeroplane and yz-plane are parabolas (Figure \(\PageIndex{ix}\)). Other elliptic paraboloids can accept other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation \( \dfrac{x^2}{a^2}+\dfrac{z^2}{c^2}=\dfrac{y}{b}\) or \( \dfrac{y^ii}{b^two}+\dfrac{z^2}{c^2}=\dfrac{x}{a}\).

Figure \(\PageIndex{9}\): This quadric surface is called an elliptic paraboloid.

Case \( \PageIndex{three}\): Identifying Traces of Quadric Surfaces

Draw the traces of the elliptic paraboloid \( ten^2+\dfrac{y^2}{2^2}=\dfrac{z}{5}\).

Solution

To detect the trace in the \(xy\)-aeroplane, fix \( z=0: x^2+\dfrac{y^ii}{two^ii}=0.\) The trace in the plane \( z=0\) is but one point, the origin. Since a single point does non tell us what the shape is, we can move up the \(z\)-centrality to an arbitrary plane to find the shape of other traces of the effigy.

The trace in plane \( z=5\) is the graph of equation \( 10^two+\dfrac{y^ii}{2^ii}=one\), which is an ellipse. In the \(xz\)-plane, the equation becomes \( z=5x^2\). The trace is a parabola in this plane and in any plane with the equation \( y=b\).

In planes parallel to the \(yz\)-plane, the traces are likewise parabolas, as we can come across in Figure \(\PageIndex{ten}\).

Figure \(\PageIndex{x}\): (a) The paraboloid \( x^two+\dfrac{y^2}{2^2}=\dfrac{z}{v}\). (b) The trace in plane \( z=5\). (c) The trace in the \(xz\)-aeroplane. (d) The trace in the \(yz\)-aeroplane.

Exercise \( \PageIndex{2}\):

A hyperboloid of one sheet is any surface that tin can be described with an equation of the form \( \dfrac{x^2}{a^ii}+\dfrac{y^ii}{b^2}−\dfrac{z^2}{c^2}=ane\). Describe the traces of the hyperboloid of i sheet given past equation \( \dfrac{ten^2}{3^2}+\dfrac{y^2}{2^2}−\dfrac{z^ii}{5^2}=1.\)

Hint

To find the traces in the coordinate planes, set each variable to zero individually.

Respond

The traces parallel to the \(xy\)-airplane are ellipses and the traces parallel to the \(xz\)- and \(yz\)-planes are hyperbolas. Specifically, the trace in the \(xy\)-plane is ellipse \( \dfrac{x^ii}{3^2}+\dfrac{y^2}{2^2}=i,\) the trace in the \(xz\)-plane is hyperbola \( \dfrac{x^two}{3^2}−\dfrac{z^two}{5^two}=1,\) and the trace in the \(yz\)-plane is hyperbola \( \dfrac{y^2}{2^2}−\dfrac{z^2}{5^2}=ane\) (see the post-obit figure).

Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure \(\PageIndex{1a}\). In fact, cooling towers for nuclear power plants are oft constructed in the shape of a hyperboloid. The builders are able to use directly steel beams in the construction, which makes the towers very strong while using relatively lilliputian cloth (Figure \(\PageIndex{1b}\)).

Figure \(\PageIndex{xi}\): (a) A sculpture in the shape of a hyperboloid can exist constructed of straight lines. (b) Cooling towers for nuclear power plants are often built in the shape of a hyperboloid.

Example \( \PageIndex{4}\): Chapter Opener: Finding the Focus of a Parabolic Reflector

Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Effigy \(\PageIndex{12}\)). If the surface of a parabolic reflector is described by equation \( \dfrac{x^two}{100}+\dfrac{y^2}{100}=\dfrac{z}{iv},\) where is the focal point of the reflector?

Figure \(\PageIndex{12}\): Energy reflects off of the parabolic reflector and is collected at the focal point. (credit: modification of CGP Grey, Wikimedia Commons)

Solution

Since z is the commencement-power variable, the axis of the reflector corresponds to the \(z\)-axis. The coefficients of \( x^2\) and \( y^2\) are equal, then the cantankerous-section of the paraboloid perpendicular to the \(z\)-centrality is a circle. We can consider a trace in the xz-plane or the yz-plane; the result is the same. Setting \( y=0\), the trace is a parabola opening upward along the \(z\)-axis, with standard equation \( ten^two=4pz\), where \( p\) is the focal length of the parabola. In this instance, this equation becomes \( 10^2=100⋅\dfrac{z}{4}=4pz\) or \( 25=4p\). So p is \( six.25\) m, which tells us that the focus of the paraboloid is \( 6.25\) thousand upward the centrality from the vertex. Because the vertex of this surface is the origin, the focal point is \( (0,0,half-dozen.25).\)

Seventeen standard quadric surfaces tin can be derived from the full general equation

\[Ax^2+By^two+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.\]

The following figures summarizes the most of import ones.

Figure \(\PageIndex{13}\): Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet, Hyperboloid of Two Sheets.

Figure \(\PageIndex{fourteen}\): Characteristics of Common Quadratic Surfaces: Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid.

Example \( \PageIndex{5}\): Identifying Equations of Quadric Surfaces

Identify the surfaces represented by the given equations.

1. \( 16x^2+9y^2+16z^2=144\)
2. \( 9x^2−18x+4y^ii+16y−36z+25=0\)

Solution

a. The \( 10,y,\) and \( z\) terms are all squared, and are all positive, then this is probably an ellipsoid. Still, allow'due south put the equation into the standard class for an ellipsoid just to be sure. Nosotros have

\[ 16x^2+9y^2+16z^2=144. \nonumber\]

Dividing through by 144 gives

\[ \dfrac{x^2}{9}+\dfrac{y^2}{16}+\dfrac{z^two}{9}=1. \nonumber\]

And then, this is, in fact, an ellipsoid, centered at the origin.

b. Nosotros first notice that the \( z\) term is raised only to the offset power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. Nosotros also note there are \( x\) terms and \( y\) terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. Nosotros have

\[ \begin{align*} 9x^2−18x+4y^ii+16y−36z+25 =0 \\[4pt] 9x^two−18x+4y^ii+16y+25 =36z \\[4pt] 9(x^ii−2x)+4(y^ii+4y)+25 =36z \\[4pt] ix(x^two−2x+1−i)+4(y^2+4y+4−4)+25 =36z \\[4pt] ix(x−1)^2−nine+four(y+2)^ii−16+25 =36z \\[4pt] 9(x−i)^2+4(y+2)^2 =36z \\[4pt] \dfrac{(x−1)^2}{4}+\dfrac{(y−2)^2}{9} =z. \finish{marshal*}\]

This is an elliptic paraboloid centered at \( (1,two,0).\)

Exercise \( \PageIndex{3}\)

Identify the surface represented by equation \( 9x^2+y^2−z^2+2z−10=0.\)

Hint

Await at the signs and powers of the \( 10,y\), and \( z\) terms

Answer

Hyperboloid of one sail, centered at \( (0,0,1)\).

Key Concepts

• A ready of lines parallel to a given line passing through a given bend is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
• The intersection of a three-dimensional surface and a plane is called a trace. To find the trace in the \(xy\)-, \(yz\)-, or \(xz\)-planes, set \( z=0,10=0,\) or \( y=0,\) respectively.
• Quadric surfaces are three-dimensional surfaces with traces equanimous of conic sections. Every quadric surface can be expressed with an equation of the form

\[Ax^ii+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+M=0. \nonumber\]

• To sketch the graph of a quadric surface, outset by sketching the traces to understand the framework of the surface.
• Important quadric surfaces are summarized in Figures \(\PageIndex{13}\) and \(\PageIndex{xiv}\).

Glossary

cylinder

a set of lines parallel to a given line passing through a given curve

ellipsoid

a three-dimensional surface described by an equation of the form \( \dfrac{x^ii}{a^2}+\dfrac{y^ii}{b^ii}+\dfrac{z^two}{c^2}=1\); all traces of this surface are ellipses

elliptic cone

a 3-dimensional surface described past an equation of the course \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^two}=0\); traces of this surface include ellipses and intersecting lines

elliptic paraboloid

a three-dimensional surface described past an equation of the class \( z=\dfrac{x^2}{a^2}+\dfrac{y^two}{b^2}\); traces of this surface include ellipses and parabolas

hyperboloid of one sheet

a 3-dimensional surface described by an equation of the form \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^two}−\dfrac{z^2}{c^two}=i;\) traces of this surface include ellipses and hyperbolas

hyperboloid of 2 sheets

a three-dimensional surface described by an equation of the form \( \dfrac{z^2}{c^2}−\dfrac{x^2}{a^2}−\dfrac{y^ii}{b^ii}=one\); traces of this surface include ellipses and hyperbolas

quadric surfaces

surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)

rulings

parallel lines that make upwardly a cylindrical surface

trace

the intersection of a three-dimensional surface with a coordinate plane

Contributors and Attributions

• Gilbert Strang (MIT) and Edwin "Jed" Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-Past-SA-NC 4.0 license. Download for complimentary at http://cnx.org.

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