Alternative Coordinate Systems and Surfaces: Learn It 4
Quadric Surfaces
We’ve studied surfaces described by first-order equations (planes). Now we’ll explore surfaces described by second-order equations, called quadric surfaces. These are 3D extensions of familiar conic sections: ellipses, parabolas, and hyperbolas.
quadric surfaces
Quadric surfaces are the graphs of equations that can be expressed in the form:
When a quadric surface intersects a coordinate plane, the trace is a conic section. Since conic sections appear frequently as traces of quadric surfaces, let’s review their equations.
Parabolas:
[latex]x^2 = 4py[/latex]: opens upward if [latex]p > 0[/latex], downward if [latex]p < 0[/latex]
[latex]y^2 = 4px[/latex]: opens rightward if [latex]p > 0[/latex], leftward if [latex]p < 0[/latex]
Ellipses:
[latex]\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1[/latex]: horizontal major axis if [latex]a > b[/latex], vertical if [latex]b > a[/latex]
Hyperbolas:
[latex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/latex]: opens horizontally (left and right)
[latex]\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1[/latex]: opens vertically (up and down)
Ellipsoids
An ellipsoid has the equation [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/latex]. Think of an ellipsoid as a 3D generalization of an ellipse. To understand its shape, examine what happens when you intersect it with each coordinate plane:
If [latex]a = b[/latex], the trace in the [latex]xy[/latex]-plane is a circle
If [latex]a = c[/latex], the trace in the [latex]xz[/latex]-plane is a circle
If [latex]b = c[/latex], the trace in the [latex]yz[/latex]-plane is a circle
If [latex]a = b = c[/latex], the ellipsoid becomes a sphere
Sketch the ellipsoid [latex]\frac{x^2}{2^2}+\frac{y^2}{3^2}+\frac{z^2}{5^2}=1[/latex].
Start by sketching the traces. To find the trace in the [latex]xy[/latex]-plane, set [latex]z=0[/latex]: [latex]\frac{x^2}{2^2}+\frac{y^2}{3^2}=1[/latex] (see Figure 1). To find the other traces, first set [latex]y=0[/latex] and then set [latex]x=0[/latex].
Figure 1. (a) This graph represents the trace of equation [latex]\frac{x^2}{2^2}+\frac{y^2}{3^2}=1[/latex] in the [latex]xy[/latex]-plane, when we set [latex]z=0[/latex]. (b) When we set [latex]y=0[/latex], we get the trace of the ellipsoid in the [latex]xz[/latex]-plane, which is an ellipse. (c) When we set [latex]x=0[/latex], we get the trace of the ellipsoid in the [latex]yz[/latex]-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 (Figure 2).
Figure 2. (a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.
The ellipsoid extends [latex]a[/latex] units along the [latex]x[/latex]-axis, [latex]b[/latex] units along the [latex]y[/latex]-axis, and [latex]c[/latex] units along the [latex]z[/latex]-axis from the center. These are called the semi-axes of the ellipsoid.
Elliptic Paraboloids
An elliptic paraboloid has equations of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}[/latex], [latex]\frac{x^2}{a^2}+\frac{z^2}{c^2}=\frac{y}{b}[/latex], or [latex]\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x}{a}[/latex]. The key feature is that one variable appears to the first power while the other two are squared. This single variable determines the axis of the paraboloid.
Unlike ellipsoids where all traces are ellipses, elliptic paraboloids have different types of traces:
Traces perpendicular to the axis: ellipses (or circles when [latex]a = b[/latex])
Traces parallel to the axis: parabolas
For [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}[/latex]:
The [latex]z[/latex]-axis is the axis of the paraboloid
Cross-sections at constant [latex]z[/latex] values are ellipses
Cross-sections in planes containing the [latex]z[/latex]-axis are parabolas
Opens upward if [latex]c > 0[/latex], downward if [latex]c < 0[/latex]
The surface looks like a bowl (or upside-down bowl) with elliptical cross-sections.
Figure 3. This quadric surface is called an elliptic paraboloid.
Other elliptic paraboloids can have different orientations by changing which variable appears to the first power:
[latex]\frac{x^2}{a^2}+\frac{z^2}{c^2}=\frac{y}{b}[/latex] (axis along [latex]y[/latex]-direction)
[latex]\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x}{a}[/latex] (axis along [latex]x[/latex]-direction)
Describe the traces of the elliptic paraboloid [latex]x^2+\frac{y^2}{2^2}=\frac{z}{5}[/latex].
To find the trace in the [latex]xy[/latex]-plane, set [latex]z=0[/latex]: [latex]x^2+\frac{y^2}{2^2}=0[/latex]. The trace in the plane [latex]z=0[/latex] is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the [latex]z[/latex]-axis to an arbitrary plane to find the shape of other traces of the figure.
The trace in plane [latex]z=5[/latex] is the graph of equation [latex]x^2+\frac{y^2}{2^2}=1[/latex], which is an ellipse. In the [latex]xz[/latex]-plane, the equation becomes [latex]z=5x^{2}[/latex]. The trace is a parabola in this plane and in any plane with the equation [latex]y=b[/latex].
In planes parallel to the [latex]yz[/latex]-plane, the traces are also parabolas, as we can see in the following figure.
Figure 4. (a) The paraboloid [latex]x^2+\frac{y^2}{2^2}=\frac{z}{5}[/latex]. (b) The trace in plane [latex]z=5[/latex]. (c) The trace in the [latex]xz[/latex]-plane. (d) The trace in the [latex]yz[/latex]-plane.
The traces tell the complete story. At [latex]z = 0[/latex], you get a single point (the vertex). As [latex]z[/latex] increases, the traces become larger ellipses. The parabolic traces show how the surface curves upward from the vertex.
Common Quadric Surfaces
The general quadric equation ([latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[/latex]) yields 17 different standard surfaces. The following figures summarizes the most important ones.
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Figure 8. Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet, Hyperboloid of Two Sheets.
Figure 9. Characteristics of Common Quadratic Surfaces: Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid.
Identification Strategy:
Look at the powers of variables (squared vs. first power)
Check the signs (positive vs. negative coefficients)
Examine traces in coordinate planes
Put equation in standard form if needed
Identify the surfaces represented by the given equations.
[latex]16x^{2}+9y^{2}+16z^{2}=144[/latex]
[latex]9x^{2}-18x+4y^{2}+16y-36z+25=0[/latex]
The [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex] terms are all squared, and are all positive, so this is probably an ellipsoid. However, let’s put the equation into the standard form for an ellipsoid just to be sure. We have
So, this is, in fact, an ellipsoid, centered at the origin.
We first notice that the [latex]z[/latex] term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are [latex]x[/latex] terms and [latex]y[/latex] 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. We have
This is an elliptic paraboloid centered at [latex](1, 2, 0)[/latex].
Real-World Applications
Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure 6(a). In fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. The builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material (Figure 6(b)).
Figure 6. (a) A sculpture in the shape of a hyperboloid can be constructed of straight lines. (b) Cooling towers for nuclear power plants are often built in the shape of a hyperboloid.
Satellite dishes and solar collectors use parabolic shapes because energy hitting the surface concentrates at the focal point.
Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure 7). If the surface of a parabolic reflector is described by equation [latex]\frac{x^2}{100}+\frac{y^2}{100}=\frac{z}{4}[/latex], where is the focal point of the reflector?
Figure 7. Energy reflects off of the parabolic reflector and is collected at the focal point. (credit: modification of CGP Grey, Wikimedia Commons)
Since [latex]z[/latex] is the first-power variable, the axis of the reflector corresponds to the [latex]z[/latex]-axis. The coefficients of [latex]x^{2}[/latex] and [latex]y^{2}[/latex] are equal, so the cross-section of the paraboloid perpendicular to the [latex]z[/latex]-axis is a circle. We can consider a trace in the [latex]xz[/latex]-plane or the [latex]yz[/latex]-plane; the result is the same. Setting [latex]y=0[/latex], the trace is a parabola opening up along the [latex]z[/latex]-axis, with standard equation [latex]x^{2}=4pz[/latex], where [latex]p[/latex] is the focal length of the parabola. In this case, this equation becomes [latex]x^2=100\cdot\frac{z}4=4pz[/latex] or [latex]25=4p[/latex]. So [latex]p[/latex] is [latex]6.25[/latex] m, which tells us that the focus of the paraboloid is [latex]6.25[/latex] m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is [latex](0, 0, 6.25)[/latex].
