Quadric Surfaces

Identifying Cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning 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-up 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 \(3.\) In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the z-axis (), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the z-axis passing through circle \({x}^{2}+{y}^{2}=9\) in the xy-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.

From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line ().

Example

Try it.

Sketch the graphs of the following cylindrical surfaces.

1. \({x}^{2}+{z}^{2}=25\)
2. \(z=2{x}^{2}-y\)
3. \(y=\text{sin}\ x\)

Solution

1. The variable \(y\) can take on any 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 circle centered at the origin with radius \(5\) (see the following figure).
   
2. In this case, the equation contains all three variables \(—x,y,\) and \(z—\) so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
   
3. In this equation, the variable z can take on any value without limit. Therefore, the lines composing this surface are parallel to the z-axis. The intersection of this surface with the xy-plane outlines curve \(y=\ \text{sin}\ x\) (see the following figure).
   

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 .

Condensed — the full section is in OpenStax Calculus Volume 3.

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.

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 \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1.\) Set \(x=0\) to see the trace of the ellipsoid in the yz-plane. To see the traces in the xy- 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 \(a=b=c.\)

Example

Try it.

Sketch the ellipsoid \(\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}+\frac{{z}^{2}}{{5}^{2}}=1.\)

Solution

Start by sketching the traces. To find the trace in the xy-plane, set \(z=0\text{:}\) \(\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}=1\) (see ). To find the other traces, first set \(y=0\) and then set \(x=0.\)

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

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 \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=\frac{z}{c},\) 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 (). 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 \(\frac{{x}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=\frac{y}{b}\) or \(\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=\frac{x}{a}.\)

Seventeen standard quadric surfaces can be derived from the general equation

\[A{x}^{2}+B{y}^{2}+C{z}^{2}+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.\]

The following figures summarizes the most important ones.

Condensed — the full section is in OpenStax Calculus Volume 3.

Key Concepts

• A set of lines parallel to a given line passing through a given curve 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 \(z=0,x=0,\ \text{or}\ y=0,\) respectively.
• Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form \(A{x}^{2}+B{y}^{2}+C{z}^{2}+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.\)
• To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
• Important quadric surfaces are summarized in and .

Quadric Surfaces

For the following exercises, sketch and describe the cylindrical surface of the given equation.

For the following exercises, the graph of a quadric surface is given.

1. Specify the name of the quadric surface.
2. Determine the axis of the quadric surface.

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

1. \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{12}=1\)
2. \(\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}-\frac{{z}^{2}}{12}=1\)
3. \(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}+\frac{{z}^{2}}{12}=1\)
4. \(z=4{x}^{2}+3{y}^{2}\)
5. \(z=4{x}^{2}-{y}^{2}\)
6. \(4{x}^{2}+{y}^{2}-{z}^{2}=0\)

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

For the following exercises, the equation of a quadric surface is given.

1. Use the method of completing the square to write the equation in standard form.
2. Identify the surface.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Sketch the graphs of the following cylindrical surfaces.

   1. \({x}^{2}+{z}^{2}=25\)
   2. \(z=2{x}^{2}-y\)
   3. \(y=\text{sin}\ x\)
   Kuratidza mhinduro

   1. The variable \(y\) can take on any 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 circle centered at the origin with radius \(5\) (see the following figure).
      
   2. In this case, the equation contains all three variables \(—x,y,\) and \(z—\) so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
      
   3. In this equation, the variable z can take on any value without limit. Therefore, the lines composing this surface are parallel to the z-axis. The intersection of this surface with the xy-plane outlines curve \(y=\ \text{sin}\ x\) (see the following figure).
      

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

   Kuratidza mhinduro

   
   

3. Sketch the ellipsoid \(\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}+\frac{{z}^{2}}{{5}^{2}}=1.\)

   Kuratidza mhinduro

   Start by sketching the traces. To find the trace in the xy-plane, set \(z=0\text{:}\) \(\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}=1\) (see ). To find the other traces, first set \(y=0\) and then set \(x=0.\)

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

4. Describe the traces of the elliptic paraboloid \({x}^{2}+\frac{{y}^{2}}{{2}^{2}}=\frac{z}{5}.\)

   Kuratidza mhinduro

   To find the trace in the xy-plane, set \(z=0\text{:}\) \({x}^{2}+\frac{{y}^{2}}{{2}^{2}}=0.\) The trace in the plane \(z=0\) is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the z-axis to an arbitrary plane to find the shape of other traces of the figure.

   The trace in plane \(z=5\) is the graph of equation \({x}^{2}+\frac{{y}^{2}}{{2}^{2}}=1,\) which is an ellipse. In the xz-plane, the equation becomes \(z=5{x}^{2}.\) 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 also parabolas, as we can see in the following figure.

5. A hyperboloid of one sheet is any surface that can be described with an equation of the form \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}=1.\) Describe the traces of the hyperboloid of one sheet given by equation \(\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}-\frac{{z}^{2}}{{5}^{2}}=1.\)

   Kuratidza mhinduro

   The traces parallel to the xy-plane are ellipses and the traces parallel to the xz- and yz-planes are hyperbolas. Specifically, the trace in the xy-plane is ellipse \(\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1,\) the trace in the xz-plane is hyperbola \(\frac{{x}^{2}}{{3}^{2}}-\frac{{z}^{2}}{{5}^{2}}=1,\) and the trace in the yz-plane is hyperbola \(\frac{{y}^{2}}{{2}^{2}}-\frac{{z}^{2}}{{5}^{2}}=1\) (see the following figure).
   

6. Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (). If the surface of a parabolic reflector is described by equation \(\frac{{x}^{2}}{100}+\frac{{y}^{2}}{100}=\frac{z}{4},\) where is the focal point of the reflector?

   Kuratidza mhinduro

   Since z is the first-power variable, the axis of the reflector corresponds to the z-axis. The coefficients of \({x}^{2}\) and \({y}^{2}\) are equal, so the cross-section of the paraboloid perpendicular to the z-axis 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 up along the z-axis, with standard equation \({x}^{2}=4pz,\) where \(p\) is the focal length of the parabola. In this case, this equation becomes \({x}^{2}=100\cdot \frac{z}{4}=4pz\) or \(25=4p.\) So p is \(6.25\) m, which tells us that the focus of the paraboloid is \(6.25\) m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is \((0,0,6.25).\)

7. Identify the surfaces represented by the given equations.

   1. \(16{x}^{2}+9{y}^{2}+16{z}^{2}=144\)
   2. \(9{x}^{2}-18x+4{y}^{2}+16y-36z+25=0\)
   Kuratidza mhinduro

   1. The \(x,y,\) and \(z\) 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
      \[16{x}^{2}+9{y}^{2}+16{z}^{2}=144.\]
      Dividing through by 144 gives
      \[\frac{{x}^{2}}{9}+\frac{{y}^{2}}{16}+\frac{{z}^{2}}{9}=1.\]
      So, this is, in fact, an ellipsoid, centered at the origin.
   2. We first notice that the \(z\) term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We 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. We have
      \[\begin{array}{lll} \\ 9{x}^{2}-18x+4{y}^{2}+16y-36z+25 & = & 0 \\ 9{x}^{2}-18x+4{y}^{2}+16y+25 & = & 36z \\ 9({x}^{2}-2x)+4({y}^{2}+4y)+25 & = & 36z \\ 9({x}^{2}-2x+1-1)+4({y}^{2}+4y+4-4)+25 & = & 36z \\ 9{(x-1)}^{2}-9+4{(y+2)}^{2}-16+25 & = & 36z \\ 9{(x-1)}^{2}+4{(y+2)}^{2} & = & 36z \\ \frac{{(x-1)}^{2}}{4}+\frac{{(y+2)}^{2}}{9} & = & z.\end{array}\]
      This is an elliptic paraboloid centered at \((1,2,0).\)

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

   Kuratidza mhinduro

   Hyperboloid of one sheet, centered at \((0,0,1)\)

9. [T] \({x}^{2}+{z}^{2}=1\)

   Kuratidza mhinduro

   The surface is a cylinder with the rulings parallel to the y-axis.
   

10. [T] \({x}^{2}+{y}^{2}=9\)

11. [T] \(z=\text{cos}(\frac{\pi }{2}+x)\)

    Kuratidza mhinduro

    The surface is a cylinder with rulings parallel to the y-axis.
    

12. [T] \(z={e}^{x}\)

13. [T] \(z=9-{y}^{2}\)

    Kuratidza mhinduro

    The surface is a cylinder with rulings parallel to the x-axis.
    

14. [T] \(z=\text{ln}(x)\)

15. Hyperboloid of two sheets

    Kuratidza mhinduro

    b.

16. Ellipsoid

17. Elliptic paraboloid

    Kuratidza mhinduro

    d.

18. Hyperbolic paraboloid

19. Hyperboloid of one sheet

    Kuratidza mhinduro

    a.

20. Elliptic cone

21. \(\text{-}{x}^{2}+36{y}^{2}+36{z}^{2}=9\)

    Kuratidza mhinduro

    \(-\frac{{x}^{2}}{9}+\frac{{y}^{2}}{\frac{1}{4}}+\frac{{z}^{2}}{\frac{1}{4}}=1,\) hyperboloid of one sheet with the x-axis as its axis of symmetry

22. \(-4{x}^{2}+25{y}^{2}+{z}^{2}=100\)

23. \(-3{x}^{2}+5{y}^{2}-{z}^{2}=10\)

    Kuratidza mhinduro

    \(-\frac{{x}^{2}}{\frac{10}{3}}+\frac{{y}^{2}}{2}-\frac{{z}^{2}}{10}=1,\) hyperboloid of two sheets with the y-axis as its axis of symmetry

24. \(3{x}^{2}-{y}^{2}-6{z}^{2}=18\)

25. \(5y={x}^{2}-{z}^{2}\)

    Kuratidza mhinduro

    \(y=-\frac{{z}^{2}}{5}+\frac{{x}^{2}}{5},\) hyperbolic paraboloid with the y-axis as its axis of symmetry

26. \(8{x}^{2}-5{y}^{2}-10z=0\)

27. \({x}^{2}+5{y}^{2}+3{z}^{2}-15=0\)

    Kuratidza mhinduro

    \(\frac{{x}^{2}}{15}+\frac{{y}^{2}}{3}+\frac{{z}^{2}}{5}=1,\) ellipsoid

28. \(63{x}^{2}+7{y}^{2}+9{z}^{2}-63=0\)

29. \({x}^{2}+5{y}^{2}-8{z}^{2}=0\)

    Kuratidza mhinduro

    \(\frac{{x}^{2}}{40}+\frac{{y}^{2}}{8}-\frac{{z}^{2}}{5}=0,\) elliptic cone with the z-axis as its axis of symmetry

30. \(5{x}^{2}-4{y}^{2}+20{z}^{2}=0\)

31. \(6x=3{y}^{2}+2{z}^{2}\)

    Kuratidza mhinduro

    \(x=\frac{{y}^{2}}{2}+\frac{{z}^{2}}{3},\) elliptic paraboloid with the x-axis as its axis of symmetry

32. \(49y={x}^{2}+7{z}^{2}\)

33. [T] \({x}^{2}+{z}^{2}+4y=0,z=0\)

    Kuratidza mhinduro

    Parabola \(y=-\frac{{x}^{2}}{4},\)
    

34. [T] \({x}^{2}+{z}^{2}+4y=0,x=0\)

35. [T] \(-4{x}^{2}+25{y}^{2}+{z}^{2}=100,x=0\)

    Kuratidza mhinduro

    Ellipse \(\frac{{y}^{2}}{4}+\frac{{z}^{2}}{100}=1,\)
    

36. [T] \(-4{x}^{2}+25{y}^{2}+{z}^{2}=100,y=0\)

37. [T] \({x}^{2}+\frac{{y}^{2}}{4}+\frac{{z}^{2}}{100}=1,x=0\)

    Kuratidza mhinduro

    Ellipse \(\frac{{y}^{2}}{4}+\frac{{z}^{2}}{100}=1,\)
    

38. [T] \({x}^{2}-y-{z}^{2}=1,y=0\)

39. Use the graph of the given quadric surface to answer the questions.

    1. Specify the name of the quadric surface.
    2. Which of the equations—\(16{x}^{2}+9{y}^{2}+36{z}^{2}=3600,9{x}^{2}+36{y}^{2}+16{z}^{2}=3600,\) or \(36{x}^{2}+9{y}^{2}+16{z}^{2}=3600\)—corresponds to the graph?
    3. Use b. to write the equation of the quadric surface in standard form.
    Kuratidza mhinduro

    a. Ellipsoid; b. The third equation; c. \(\frac{{x}^{2}}{100}+\frac{{y}^{2}}{400}+\frac{{z}^{2}}{225}=1\)

40. Use the graph of the given quadric surface to answer the questions.

    1. Specify the name of the quadric surface.
    2. Which of the equations—\(36z=9{x}^{2}+{y}^{2},9{x}^{2}+4{y}^{2}=36z,\ \text{or}\ -36z=-81{x}^{2}+4{y}^{2}\)—corresponds to the graph above?
    3. Use b. to write the equation of the quadric surface in standard form.

Symbols used here

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\frac{\partial f}{\partial x}

partial derivative
Derivative with respect to x, holding the other variables fixed.

\nabla f

gradient (nabla, del)
Vector of partial derivatives; points uphill.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

\iint,\ \oint

double / contour integral
Integral over a region of the plane; integral around a closed curve.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|

dot product, norm
Σ u_i v_i; the length of v, √(v·v).

How to: Quadric Surfaces

1. Identify a cylinder as a type of three-dimensional surface.
2. Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
3. Use traces to draw the intersections of quadric surfaces with the coordinate planes.
4. The variable
5. In this case, the equation contains all three variables
6. In this equation, the variable
7. The
8. We first notice that the

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen — the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

Tarisa yako

⌨ Type ✎ Draw

Yako maths nyaya

_Kugadzirisa

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Calculus Volume 3 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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