Definition10.1.3Distance In Space
Let \(P=(x_1,y_1,z_1)\) and \(Q = (x_2,y_2,z_2)\) be points in space. The distance \(D\) between \(P\) and \(Q\) is \begin{equation*} D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. \end{equation*}
Up to this point in this text we have considered mathematics in a 2–dimensional world. We have plotted graphs on the \(x\)-\(y\) plane using rectangular and polar coordinates and found the area of regions in the plane. We have considered properties of solid objects, such as volume and surface area, but only by first defining a curve in the plane and then rotating it out of the plane.
While there is wonderful mathematics to explore in “2D,” we live in a “3D” world and eventually we will want to apply mathematics involving this third dimension. In this section we introduce Cartesian coordinates in space and explore basic surfaces. This will lay a foundation for much of what we do in the remainder of the text.
Each point \(P\) in space can be represented with an ordered triple, \(P=(a,b,c)\text{,}\) where \(a\text{,}\) \(b\) and \(c\) represent the relative position of \(P\) along the \(x\)-, \(y\)- and \(z\)-axes, respectively. Each axis is perpendicular to the other two.
Visualizing points in space on paper can be problematic, as we are trying to represent a 3-dimensional concept on a 2–dimensional medium. We cannot draw three lines representing the three axes in which each line is perpendicular to the other two. Despite this issue, standard conventions exist for plotting shapes in space that we will discuss that are more than adequate.
One convention is that the axes must conform to the right hand rule. This rule states that when the index finger of the right hand is extended in the direction of the positive \(x\)-axis, and the middle finger (bent “inward” so it is perpendicular to the palm) points along the positive \(y\)-axis, then the extended thumb will point in the direction of the positive \(z\)-axis. (It may take some thought to verify this, but this system is inherently different from the one created by using the “left hand rule.”)
As long as the coordinate axes are positioned so that they follow this rule, it does not matter how the axes are drawn on paper. There are two popular methods that we briefly discuss.
In Figure 10.1.1 we see the point \(P=(2,1,3)\) plotted on a set of axes. The basic convention here is that the \(x\)-\(y\) plane is drawn in its standard way, with the \(z\)-axis down to the left. The perspective is that the paper represents the \(x\)-\(y\) plane and the positive \(z\) axis is coming up, off the page. This method is preferred by many engineers. Because it can be hard to tell where a single point lies in relation to all the axes, dashed lines have been added to let one see how far along each axis the point lies.
One can also consider the \(x\)-\(y\) plane as being a horizontal plane in, say, a room, where the positive \(z\)-axis is pointing up. When one steps back and looks at this room, one might draw the axes as shown in Figure 10.1.2. The same point \(P\) is drawn, again with dashed lines. This point of view is preferred by most mathematicians, and is the convention adopted by this text.
It is of critical importance to know how to measure distances between points in space. The formula for doing so is based on measuring distance in the plane, and is known (in both contexts) as the Euclidean measure of distance.
Let \(P=(x_1,y_1,z_1)\) and \(Q = (x_2,y_2,z_2)\) be points in space. The distance \(D\) between \(P\) and \(Q\) is \begin{equation*} D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. \end{equation*}
We refer to the line segment that connects points \(P\) and \(Q\) in space as \(\overline{PQ}\text{,}\) and refer to the length of this segment as \(\norm{\overline{PQ}}\text{.}\) The above distance formula allows us to compute the length of this segment.
Let \(P = (1,4,-1)\) and let \(Q = (2,1,1)\text{.}\) Draw the line segment \(\overline{PQ}\) and find its length.
Just as a circle is the set of all points in the plane equidistant from a given point (its center), a sphere is the set of all points in space that are equidistant from a given point. Definition 10.1.3 allows us to write an equation of the sphere.
We start with a point \(C = (a,b,c)\) which is to be the center of a sphere with radius \(r\text{.}\) If a point \(P=(x,y,z)\) lies on the sphere, then \(P\) is \(r\) units from \(C\text{;}\) that is, \begin{equation*} \norm{\overline{PC}} = \sqrt{(x-a)^2+(y-b)^2+(z-c)^2} = r. \end{equation*}
Squaring both sides, we get the standard equation of a sphere in space with center at \(C=(a,b,c)\) with radius \(r\text{,}\) as given in the following Key Idea.
The standard equation of the sphere with radius \(r\text{,}\) centered at \(C=(a,b,c)\text{,}\) is \begin{equation*} (x-a)^2+(y-b)^2+(z-c)^2=r^2. \end{equation*}
Find the center and radius of the sphere defined by \(x^2+2x+y^2-4y+z^2-6z=2\text{.}\)
The equation of a sphere is an example of an implicit function defining a surface in space. In the case of a sphere, the variables \(x\text{,}\) \(y\) and \(z\) are all used. We now consider situations where surfaces are defined where one or two of these variables are absent.
The coordinate axes naturally define three planes (shown in Figure 10.1.3), the coordinate planes: the \(x\)-\(y\) plane, the \(y\)-\(z\) plane and the \(x\)-\(z\) plane. The \(x\)-\(y\) plane is characterized as the set of all points in space where the \(z\)-value is 0. This, in fact, gives us an equation that describes this plane: \(z=0\text{.}\) Likewise, the \(x\)-\(z\) plane is all points where the \(y\)-value is 0, characterized by \(y=0\text{.}\)
The equation \(x=2\) describes all points in space where the \(x\)-value is 2. This is a plane, parallel to the \(y\)-\(z\) coordinate plane, shown in Figure 10.1.9.
Sketch the region defined by the inequalities \(-1\leq y\leq 2\text{.}\)
The equation \(x=1\) obviously lacks the \(y\) and \(z\) variables, meaning it defines points where the \(y\) and \(z\) coordinates can take on any value. Now consider the equation \(x^2+y^2=1\) in space. In the plane, this equation describes a circle of radius 1, centered at the origin. In space, the \(z\) coordinate is not specified, meaning it can take on any value. In Figure 10.1.12 (a), we show part of the graph of the equation \(x^2+y^2=1\) by sketching 3 circles: the bottom one has a constant \(z\)-value of \(-1.5\text{,}\) the middle one has a \(z\)-value of 0 and the top circle has a \(z\)-value of 1. By plotting all possible \(z\)-values, we get the surface shown in Figure 10.1.12 (b). This surface looks like a “tube,” or a “cylinder”; mathematicians call this surface a cylinder for an entirely different reason.
Let \(C\) be a curve in a plane and let \(L\) be a line not parallel to \(C\text{.}\) A cylinder is the set of all lines parallel to \(L\) that pass through \(C\text{.}\) The curve \(C\) is the directrix of the cylinder, and the lines are the rulings.
In this text, we consider curves \(C\) that lie in planes parallel to one of the coordinate planes, and lines \(L\) that are perpendicular to these planes, forming right cylinders. Thus the directrix can be defined using equations involving 2 variables, and the rulings will be parallel to the axis of the 3\(^\text{ rd }\) variable.
In the example preceding the definition, the curve \(x^2+y^2=1\) in the \(x\)-\(y\) plane is the directrix and the rulings are lines parallel to the \(z\)-axis. (Any circle shown in Figure 10.1.12 can be considered a directrix; we simply choose the one where \(z=0\text{.}\)) Sample rulings can also be viewed in part (b) of the figure. More examples will help us understand this definition.
Graph the cylinder following cylinders.
1.\(z=y^2\) 2.\(x=\sin(z)\)
One of the applications of integration we learned previously was to find the volume of solids of revolution — solids formed by revolving a curve about a horizontal or vertical axis. We now consider how to find the equation of the surface of such a solid.
Consider the surface formed by revolving \(y=\sqrt{x}\) about the \(x\)-axis. Cross–sections of this surface parallel to the \(y\)-\(z\) plane are circles, as shown in Figure 10.1.19(a). Each circle has equation of the form \(y^2+z^2=r^2\) for some radius \(r\text{.}\) The radius is a function of \(x\text{;}\) in fact, it is \(r(x) = \sqrt{x}\text{.}\) Thus the equation of the surface shown in Figure 10.1.19b is \(y^2+z^2=(\sqrt{x})^2\text{.}\)
We generalize the above principles to give the equations of surfaces formed by revolving curves about the coordinate axes.
Let \(r\) be a radius function.
The equation of the surface formed by revolving \(y=r(x)\) or \(z=r(x)\) about the \(x\)-axis is \(y^2+z^2=r(x)^2\text{.}\)
The equation of the surface formed by revolving \(x=r(y)\) or \(z=r(y)\) about the \(y\)-axis is \(x^2+z^2=r(y)^2\text{.}\)
The equation of the surface formed by revolving \(x=r(z)\) or \(y=r(z)\) about the \(z\)-axis is \(x^2+y^2=r(z)^2\text{.}\)
Let \(y=\sin(z)\) on \([0,\pi]\text{.}\) Find the equation of the surface of revolution formed by revolving \(y=\sin(z)\) about the \(z\)-axis.
This particular method of creating surfaces of revolution is limited. For instance, in Example 7.3.18 of Section 7.3 we found the volume of the solid formed by revolving \(y=\sin(x)\) about the \(y\)-axis. Our current method of forming surfaces can only rotate \(y=\sin(x)\) about the \(x\)-axis. Trying to rewrite \(y=\sin(x)\) as a function of \(y\) is not trivial, as simply writing \(x=\sin^{-1}(y)\) only gives part of the region we desire.
What we desire is a way of writing the surface of revolution formed by rotating \(y=f(x)\) about the \(y\)-axis. We start by first recognizing this surface is the same as revolving \(z=f(x)\) about the \(z\)-axis. This will give us a more natural way of viewing the surface.
A value of \(x\) is a measurement of distance from the \(z\)-axis. At the distance \(r\text{,}\) we plot a \(z\)-height of \(f(r)\text{.}\) When rotating \(f(x)\) about the \(z\)-axis, we want all points a distance of \(r\) from the \(z\)-axis in the \(x\)-\(y\) plane to have a \(z\)-height of \(f(r)\text{.}\) All such points satisfy the equation \(r^2=x^2+y^2\text{;}\) hence \(r=\sqrt{x^2+y^2}\text{.}\) Replacing \(r\) with \(\sqrt{x^2+y^2}\) in \(f(r)\) gives \(z=f(\sqrt{x^2+y^2})\text{.}\) This is the equation of the surface.
Let \(z=f(x)\text{,}\) \(x\geq 0\text{,}\) be a curve in the \(x\)-\(z\) plane. The surface formed by revolving this curve about the \(z\)-axis has equation \(z=f\big(\sqrt{x^2+y^2}\big)\text{.}\)
Find the equation of the surface found by revolving \(z=\sin(x)\) about the \(z\)-axis.
Spheres, planes and cylinders are important surfaces to understand. We now consider one last type of surface, a quadric surface. The definition may look intimidating, but we will show how to analyze these surfaces in an illuminating way.
A quadric surface is the graph of the general second–degree equation in three variables: \begin{equation*} Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0. \end{equation*}
When the coefficients \(D\text{,}\) \(E\) or \(F\) are not zero, the basic shapes of the quadric surfaces are rotated in space. We will focus on quadric surfaces where these coeffiecients are 0; we will not consider rotations. There are six basic quadric surfaces: the elliptic paraboloid, elliptic cone, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, and the hyperbolic paraboloid.
We study each shape by considering traces, that is, intersections of each surface with a plane parallel to a coordinate plane. For instance, consider the elliptic paraboloid \(z= x^2/4+y^2\text{,}\) shown in Figure 10.1.33. If we intersect this shape with the plane \(z=d\) (i.e., replace \(z\) with \(d\)), we have the equation: \begin{align*} d \amp = \frac{x^2}4+y^2.\\ \end{align*} Divide both sides by \(d\text{:}\) \begin{align*} 1 \amp = \frac{x^2}{4d} + \frac{y^2}{d}. \end{align*}
This describes an ellipse — so cross sections parallel to the \(x\)-\(y\) coordinate plane are ellipses. This ellipse is drawn in the figure.
Now consider cross sections parallel to the \(x\)-\(z\) plane. For instance, letting \(y=0\) gives the equation \(z=x^2/4\text{,}\) clearly a parabola. Intersecting with the plane \(x=0\) gives a cross section defined by \(z=y^2\text{,}\) another parabola. These parabolas are also sketched in the figure.
Thus we see where the elliptic paraboloid gets its name: some cross sections are ellipses, and others are parabolas.
Such an analysis can be made with each of the quadric surfaces. We give a sample equation of each, provide a sketch with representative traces, and describe these traces.
Elliptic Paraboloid, \(\ds z=\frac{x^2}{a^2}+\frac{y^2}{b^2}\)
Plane | Trace | |
\(x=d\) | Parabola | |
\(y=d\) | Parabola | |
\(z=d\) | Ellipse |
\rule{1.1\linewidth}{.5pt}
Elliptic Cone, \(\ds z^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}\)
Plane | Trace | |
\(x=0\) | Crossed Lines | |
\(y=0\) | Crossed Lines | |
\(x=d\) | Hyperbola | |
\(y=d\) | Hyperbola | |
\(z=d\) | Ellipse |
Ellipsoid, \(\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
Plane | Trace | |
\(x=d\) | Ellipse | |
\(y=d\) | Ellipse | |
\(z=d\) | Ellipse |
\rule{1.1\linewidth}{.5pt}
Hyperboloid of One Sheet, \(\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
Plane | Trace | |
\(x=d\) | Hyperbola | |
\(y=d\) | Hyperbola | |
\(z=d\) | Ellipse |
Hyperboloid of Two Sheets, \(\ds \frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Plane | Trace | |
\(x=d\) | Hyperbola | |
\(y=d\) | Hyperbola | |
\(z=d\) | Ellipse |
\rule{1.1\linewidth}{.5pt}
Hyperbolic Paraboloid, \(\ds z=\frac{x^2}{a^2}-\frac{y^2}{b^2}\)
Plane | Trace | |
\(x=d\) | Parabola | |
\(y=d\) | Parabola | |
\(z=d\) | Hyperbola |
Sketch the quadric surface defined by the given equation.
1. \(\ds y=\frac{x^2}{4}+\frac{z^2}{16}\) 2. \(\ds x^2+\frac{y^2}{9}+\frac{z^2}{4}=1\text{.}\) 3. \(\ds z=y^2-x^2\text{.}\)
Consider the quadric surface shown in Figure 10.1.45. Which of the following equations best fits this surface?
(a) | \(\ds x^2-y^2-\frac{z^2}{9}=0\) | (c) | \(\ds z^2-x^2-y^2=1\) | |
(b) | \(\ds x^2-y^2-z^2=1\) | (d) | \(\ds 4x^2-y^2-\frac{z^2}9=1\) |
This section has introduced points in space and shown how equations can describe surfaces. The next sections explore vectors, an important mathematical object that we'll use to explore curves in space.
Terms and Concepts
In the following exercises, describe the region in space defined by the inequalities.
In the following exercises, sketch the cylinder in space.
In the following exercises, give the equation of the surface of revolution described.
In the following exercises, a quadric surface is sketched. Determine which of the given equations best fits the graph.
In the following exercises, sketch the quadric surface.