What surfaces arise as graphs of an equations that contain all three of the variables \(x\text{,}\)\(y\text{,}\) and \(z\) and are quadratic in at least one variable?
In this section, we will introduce some examples of graphs in three dimensions that have nice algebraic properties and a variety of interesting geometric features. In the Preview Activity, we ask you to practice identifying conic sections such as circles, parabolas, ellipses, and hyperbolas from their equations. The section will then help you see how to use your knowledge of conic sections to identify key properties of graphs in three dimensions. For a complete discussion of conic sections, see Section A.1.
For each equation below, identify its graph in Figure 9.7.1. Note that there are more graphs than equations, so some graphs will not be selected. To help you with identification, you might consider looking for \(x\)-intercepts, \(y\)-intercepts, and values of a variable for which the graph contains no points.
The Preview Activity gave you an opportunity recall that there are a number of interesting curves in the \(xy\)-plane that have nice equations. What happens if we consider these same equations in three dimensions?
We start by considering \((x-1)=\frac{(y+2)^2}{2}\text{.}\) The graph of \((x-1)=\frac{(y+2)^2}{2}\) in two dimensions is a parabola centered at \((1,-2)\text{.}\)
A plot of \((x-1)=\frac{(y+2)^2}{2}\) in the \(xy\)-plane. The parabola opens to the right and has vertex at \((1,-2)\text{.}\) Four additional points (two with smaller \(y\)-coordinate than the vertex and two with larger \(y\)-coordinate than the vertex) are marked on the graph.
If we want to consider the graph of \((x-1)=\frac{(y+2)^2}{2}\) in three dimensions, then the graph must include all of the \((x,y,z)\) points that will satisfy this equation. For any \(x\) and \(y\) values we pick that satisfy \((x-1)=\frac{(y+2)^2}{2}\text{,}\) we can make any choice of \(z\) to determine a point that also satisfies the given question. Hence, for the each of the highlighted points on Figure 9.7.3, we can extend the graph of \((x-1)=\frac{(y+2)^2}{2}\) parallel to the \(z\)-axis, forming a vertical line. Figure 9.7.4 shows how the points on the parabola (in black) can be extended to include any \(z\)-coordinate.
Extending all points from the parabola \((x-1)=\frac{(y+2)^2}{2}\) in the \(xy\)-plane parallel to the \(z\)-axis will give a surface. This kind of surface is called a cylinder surface. We call the two-dimensional curve used to make the surface the generating curve, and the lines that extend in the direction of the missing variable are called rulings. In Figure 9.7.5, the surface is plotted in blue, the generating curve in black, and the rulings in green. This surface is called a parabolic cylinder surface because the generating curve is a parabola.
The next activity prompts you to look at the most important cylinder surfaces. The conic sections that were reviewed in Preview Activity 9.7.1 will be helpful as you do this.
Each equation below is a cylinder surface in \(\R^3\text{.}\) To sketch the cylinder surfaces in \(xyz\)-space, you should first draw the generating curve in the \(xy\)-plane, \(xz\)-plane, or \(yz\)-plane (depending on which two variables appear in the equation) and then then sketch a three-dimensional cylinder surface by thinking about how the rulings will run.
The defining characteristic of the equations of cylinder surfaces is that one variable is completely omitted from the equation. Now we will examine surfaces with algebraic equations that are quadratic in \(x\text{,}\)\(y\text{,}\) and \(z\text{.}\) This will give us a category of example surfaces that are simple algebraically but exhibit a variety of interesting and important characteristics. To understand these quadric surfaces, we focus on the intercepts and two-dimensional graphs formed by the intersection of the quadric surface with fundamental planes. Next activity leads you through this process to help you learn to recognize and sketch quadric surfaces. After this, we will summarize the key ideas for recognizing and sketching quadric surfaces and then you will have the opportunity to apply those key ideas in additional activities.
For this activity, we will be looking at a variety properties that will help us draw a graph of the surface described by \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1\text{.}\)
Find an equation for the curve given by the intersection of \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1\) with the \(xy\)-plane, the \(yz\)-plane, and the \(xz\)-plane. Draw a two-dimensional plot of each intersection.
Find equations for the curve given by the intersection of \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1\) with the each of the following fundamental planes. You should state the shape and any other characteristics (like center or direction) for each of these intersections.
Which of the following surface plots will correspond to \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1\text{?}\) You can determine this by comparing the features on your previous part to these options.
In designing Activity 9.7.3, we carefully chose the fundamental planes for which we asked you to find the intersection with the surface. By examining the equation that defines a surface, you can strategically select fundamental planes to investigate. Doing so will allow you to identify key characteristics of a quadric surface without needing to test an exceedingly large number of fundamental planes. In Key Idea 9.7.8, we suggest some steps to follow. Afterward, you will have an activity that gives you a chance to practice implementing the steps.
Identify the intercepts of the surface with the \(x\)-axis, \(y\)-axis, and \(z\)-axis, including noting if the surface does not intersect one or more of these coordinate axes. To find the intercepts, set two of the variables equal to \(0\) and solve for the third.
Identify the intersection curve of the surface with the \(xy\)-plane, \(yz\)-plane, and \(xz\)-plane, including noting if the surface does not intersect one or more of these coordinate planes. To find these intersections, set one variable equal to \(0\) and manipulate the equation so that you can recognize the shape of the curve. In most cases, the curve will be a conic section, and you should pay particular attention to the intercepts of the conic section.
Identify the intersection curve of the surface with at least two fundamental planes parallel to each of the three coordinate planes. To find these intersections, set a variable equal to a nonzero constant and manipulate the equation so that you can recognize the shape of the curve. In most cases, the curve will be a conic section, and you should pay particular attention to the intercepts of the conic section.
Practice will help you develop a strategy for choosing these constants, but the following suggestions will be helpful as you are practicing:
Use what you learned in the two prior steps. For example, if a surface doesn’t intersect the \(xy\)-plane (\(z=0\)), find values of \(z\) that do create intersections with fundamental planes.
Determine if the shape of the intersection varies depending on the value you choose for a fixed coordinate. For example, if all intersections with planes of the form \(x=c\) are hyperbolas, do some values of \(c\) cause the hyperbolas to have intercepts on one axis while other values of \(c\) cause the hyperbolas to have intercepts on another axis? If there variations of this form, identify them as best as you can.
To illustrate parts of Key Idea 9.7.8, we can return to Activity 9.7.3 and discuss some of our strategy in selecting the fundamental planes we asked you to consider:
For planes of the form \(z=c\text{,}\) the intersection equation is always of the form \(\frac{x^2}{4}+\frac{y^2}{9} = 1+c^2\text{,}\) which gives an ellipse for all values of \(c\text{.}\)
For planes of the form \(x=c\text{,}\) the intersection equation is \(\frac{y^2}{9}-z^2 = 1-\frac{c^2}{4}\text{.}\) When \(c^2/4 \lt 1, \text{,}\) this is a hyperbola with positive coefficient on the \(y^2\) term. This hyperbola has intercepts on the \(y\)-axis. When \(c^2/4\gt 1\text{,}\) the right-hand side of the equation is negative, so a more standard form for the equation would be \(z^2-\frac{y^2}{9} = \frac{c^2}{4}-1\text{,}\) which is a hyperbola that has intercepts on the \(z\)-axis.
For planes of the form \(y=c\text{,}\) the intersection equation is \(\frac{x^2}{4}-z^2 = 1-\frac{c^2}{9}\text{.}\) As with planes of the form \(x=c\text{,}\) these are hyperbolas where the location of the intercepts depends on the sign of \(1-\frac{c^2}{9}\text{.}\)
For each equation below, use the process in Key Idea 9.7.8 to determine the shape of the quadric surface defined by the equation and identify the graph of the quadric surface in Figure 9.7.9.
As mentioned before, the surfaces given by \(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{1}=1\text{,}\)\(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1\text{,}\)\(\frac{x^2}{4}-\frac{y^2}{9}-\frac{z^2}{1}=1\text{,}\) and \(\frac{x^2}{4}-\frac{y^2}{9}-\frac{z}{1}=1\) are examples of quadric surfaces. Quadric surfaces are the surfaces generated by polynomials that are quadratic in at least one of the three coordinate variables: \(x\text{,}\)\(y\text{,}\) and \(z\text{.}\) There are six main shapes of quadric surfaces. The box below gives the forms of the equations of these. Figure 9.7.10 shows plots of each of the main shapes with the algebraic form used to describe each shape. Note that all of these examples are oriented along the \(z\)-axis and centered at the origin. However, as with conic sections, quadric surfaces can be oriented along other coordinate directions and/or centered away from the origin.
The following is a list of the six types of quadric surfaces and their general algebraic form centered at the origin with stretches of \(a\text{,}\)\(b\text{,}\) and \(c\) in the respective coordinate directions. Each of these is oriented along the \(z\)-axes (when possible).
In Chapter 11, we will talk about surfaces where one coordinate can be expressed as a function of the others. This is an extension of the idea that graphs that pass the vertical line test can be expressed with \(y\) as a function of \(x\text{.}\) For instance, in the plot of the saddle surface shown in Figure 9.7.10, any line parallel to the \(z\)-axis will intersect the saddle surface at only one place. This means that the saddle surface can be expressed with \(z\) as a function of \(x\) and \(y\text{.}\) This is not surprising since the saddle surface is given by \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=z\text{.}\)
Compare this to the graph of a hyperboloid of one sheet in Figure 9.7.10, where a line that is parallel to the \(z\)-axis will intersect the surface at two places. Lines parallel to the \(x\)- or \(y\)-axes will also intersect the hyperboloid of one sheet in more than one place. This means that we cannot express the graph of a hyperboloid of one sheet with one of the coordinates as a function of the other two. Algebraically, this corresponds to there being more than one solution when you try to solve the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) for one variable. In particular, the multiple solutions come from needing to consider the positive and negative square roots. We will return to this idea in Chapter 11.
Hyperbola: \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\) with center \((h,k)\text{,}\) vertices \((h\pm a,k)\text{,}\) and asymptotes \((y-k)=\frac{b}{a}(x-h)\)
Cylinder surfaces are described algebraically by an equation involving only two coordinate variables. Geometrically, a cylinder surface is generated by a curve/graph in the coordinate plane involving the two coordinate variables in the equation and stretching this generating curve parallel to the missing coordinate variable’s axis.
Quadric surfaces are a category of surfaces created by quadratic equations in \(x\text{,}\)\(y\text{,}\) and \(z\text{.}\) Quadric surfaces have six typical shapes: ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic paraboloid, hyperbolic paraboloid (saddle surface), and cone. The same coordinate transformations that generalize conic sections can be applied to quadric surfaces.
For each surface, decide whether it could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive z-axis is up.
