How to Draw a Hyperbolic Paraboloid - Detailed...

How to Draw a Hyperbolic ParaboloidDetailed Guide

John Ganci1 Al Lehnen2

1Richland CollegeDallas, TX

[email protected]

2Madison Area Technical CollegeMadison, WI

[email protected]

August 24, 2011

Introduction

Drawing graphs of the quadric surfaces is fairly straightforward forparaboloids, ellipsoids, and hyperboloids. However, drawing thegraph of a hyperbolic paraboloid requires some thought.

We describe a set of steps that make drawing the graph of ahyperbolic paraboloid a routine task.

Throughout the presentation we assume that we have an equationin 3 variables in the form

Au + Bv2 + Cw2 = 0

where A, B, and C are non-zero constants, A and B have thesame sign (that is, AB > 0), C is of opposite sign (that is,AC < 0), and u, v , w is some permutation of x , y , z .

Note: All coordinates in what follows are assumed to beuvw -coordinates.

The steps

If we have an equation that satisfies the conditions described onthe previous slide, then we know that its graph is a hyperbolicparaboloid. We can draw a rough sketch of the graph by followingthese steps:

Write the equation in standard form

Identify the axis

Identify two parabolas

Draw the parabolas

Identify two hyperbolas

Draw the hyperbolas

Connect the hyperbolas

Shade the surface

Write in standard form

First write the equation in standard form, which means in the form

u =v2

a2− w2

b2

where a and b are positive constants.

Examples.

1 The equation 400z + 25x2 − 16y2 = 0 can be rewritten as

z =y2

25− x2

16=

y2

52− x2

42

so u = z , v = y , w = x , a = 5, and b = 4.

2 The equation 400z + 25x2 + 16y2 = 0 cannot be written instandard form because neither of the coefficients of the degree2 terms has sign opposite the sign of the degree 1 term.

3 The equation z2 + x − y2 = 0 can be rewritten asx = y2 − z2. Here u = x , v = y , w = z , a = 1 = b.

Identify the axis

The axis of a hyperbolic paraboloid is one of the three coordinateaxes. That is, it is either the x-axis, the y -axis, or the z-axis. Oncethe equation has been written in standard form, identifying theaxis amounts to identifying the variable of degree 1. In our case,the axis is the u-axis since u is the variable of degree 1.

Examples.

1 On the previous slide we wrote the equation

400z + 25x2 − 16y2 = 0 as z = y2

52− x2

42, so the axis is the

z-axis.

2 The equation x = y2 − z2 is already in standard form. Theaxis is the x-axis.

Comment. Since there is only one variable of degree 1, the readermight question why we write the equation in standard form. Thereasons why will become apparent on subsequent slides.

Identify the parabolas (1/3)

A hyperbolic paraboloid is, roughly speaking, a surface that ismade up of hyperbolas whose vertices lie on one of two parabolas.Our next step identifies the two parabolas.

Each parabola arises from setting one of the degree 2 variables tozero. One of the parabolas, the upper parabola, opens along thepositive u-axis. The other parabola, the lower parabola, opensalong the negative u-axis.

Equation in standard form: u = v2

a2− w2

b2

The upper parabola: u = v2

a2

The lower parabola: u = −w2

b2

Example. For x = y2 − z2 the upper parabola is x = y2 and thelower parabola is x = −z2.


Each of the parabolas has its vertex at the origin and is symmetricwith respect to the u-axis.

The upper parabola u = v2

a2lies in the uv -plane.

The lower parabola u = −w2

b2lies in the uw -plane.

Example. For x = y2 − z2 the upper parabola x = y2 lies in thexy -plane. The lower parabola x = −z2 lies in the xz-plane.


We next want to draw arcs for each parabola, each arc beingsymmetric with respect to the u-axis. We do this by choosing apositive number s and then finding the values of the degree 2variable that satisfy 0 ≤ u = v2

a2≤ s2 for the upper parabola and

0 ≥ u = −w2

b2≥ −s2 for the lower parabola.

The u-coordinates of the endpoints of the arc of the upperparabola are s2. The u-coordinates of the endpoints of the arc ofthe lower parabola are −s2.

Example. For x = y2 − z2 we use s = 2. For the upper parabolax = y2 we want 0 ≤ x = y2 ≤ 22 = 4 so −2 ≤ y ≤ 2. For thelower parabola x = −z2 we want 0 ≥ x = −z2 ≥ −22 = −4 so−2 ≤ z ≤ 2.

Draw the parabolas (1/2)

Draw the upper parabola: x = y2

x

y

z

−2 ≤ y ≤ 20 ≤ x ≤ 4 Note the upper bound for x

Draw the parabolas (2/2)

Draw the lower parabola: x = −z2

x

y

z

−2 ≤ z ≤ 2Note the lower bound for x −4 ≤ x ≤ 0

Identify two hyperbolas (1/3)

We next associate a hyperbola with each parabola. The hyperbolaassociated with the upper parabola is called the upper hyperbolaand the hyperbola associated with the lower parabola is called thelower hyperbola.

The upper hyperbola lies in the plane u = s2, where s is the valuechosen for the arcs of the parabolas. Its vertices are the twoendpoints of the upper parabola. The upper hyperbola is theintersection of the hyperbolic cylinder s2 = v2

a2− w2

b2and the

plane u = s2. Note that 1 = v2

(sa)2− w2

(sb)2is an equivalent equation

for the hyperbolic cylinder.

The lower hyperbola lies in the plane u = −s2. Its vertices are thetwo endpoints of the lower parabola. The lower hyperbola is theintersection of the hyperbolic cylinder −s2 = v2

a2− w2

b2and the

plane u = −s2. Note that 1 = w2

(sb)2− v2

(sa)2is an equivalent

equation for the hyperbolic cylinder.


Since the hyperbolic cylinder associated with the upper hyperbolais symmetric with respect to both the uv -plane and the uw -plane,the upper hyperbola is symmetric with respect to two lines: thefirst is the intersection of the plane x = s2 and the uv -plane; thesecond is the intersection of the plane x = s2 and the uw -plane.Similarly, the lower hyperbola is symmetric with respect to twolines: the first is the intersection of the plane x = −s2 and theuv -plane; the second is the intersection plane x = −s2 and theuw -plane.

We next want to draw arcs for both hyperbolas. Findingappropriate bounds for the arcs is a little more complicated than itwas for the parabolas. We illustrate what to do here with a specificexample. Since we will only be drawing rough sketches ofhyperbolic paraboloids, having exact values is not necessary. Theoptional material in Appendix A contains most of the details. SeeAppendix B for references to all the details.


Example. For the equation x = y2 − z2 the two parabolas are

Upper: x = y2, −2 ≤ y ≤ 2, 0 ≤ x = y2 ≤ 22 = 4

Lower: x = −z2, −2 ≤ z ≤ 2, 0 ≥ x = −z2 ≥ −22 = −4

Each hyperbola consists of two arcs: an upper arc and a lower arc.

Upper hyperbola: 4 = y2 − z2 or 1 = y2

4 −z2

4

Upper arc: y =√

4 + z2, −2 ≤ z ≤ 2, 2 ≤ y ≤ 2√

2Lower arc: y = −

√4 + z2, −2 ≤ z ≤ 2, −2

√2 ≤ y ≤ −2

Lower hyperbola: −4 = y2 − z2 or 1 = z2

4 −y2

4

Upper arc: z =√

4 + y2, −2 ≤ y ≤ 2, 2 ≤ z ≤ 2√

2

Lower arc: z = −√

4 + y2, −2 ≤ y ≤ 2, −2√

2 ≤ z ≤ −2

The upper hyperbola lies in the plane x = 4 and the lowerhyperbola lies in the plane x = −4.

Draw the hyperbolas (1/2)

Draw the upper hyperbola: 4 = y2 − z2 or 1 = y2

4 −z2

4

x

y

z

Plane: x = 4−2 ≤ z ≤ 2

−2√

2 ≤ y ≤ −2 or 2 ≤ y ≤ 2√

2

Draw the hyperbolas (2/2)

Draw the lower hyperbola: −4 = y2 − z2 or 1 = z2

4 −y2

4

x

y

z

Plane: x = −4−2 ≤ y ≤ 2

−2√

2 ≤ z ≤ −2 or 2 ≤ z ≤ 2√

2

Connect the hyperbolas

Now we connect the hyperbolas. We do this by drawing four linesegments.

1 Connect the upper hyperbola, upper ends, to the lowerhyperbola, upper ends.

2 Connect the upper hyperbola, lower ends, to the lowerhyperbola, lower ends.

If the arcs of the two hyperbolas are appropriately matched (seethe document An Interesting Property of Hyperbolic Paraboloids)then these line segments lie on the surface of the hyperbolicparabolid. (See Appendix B for the location of the document.)

Connect the hyperbolas (1/2)

The pink line segments connect the upper ends.

x

y

z

Connect the hyperbolas (2/2)

The aqua line segments connect the lower ends.

x

y

z

Shade the surface

Draw additional hyperbolas (shown in blue) along the upperparabola, each parallel to the fixed upper hyperbola. Do the same(shown in green) for the lower parabola.

x

y

z

Shaded surface with Winplot graph

x

y

z

Summary

We have described a “recipe” you can follow to draw a roughsketch of a hyperbolic paraboloid. The mathematical details wereintentionally omitted. The interested reader should consult theappendices for the details.

Appendix

There are two appendices.

Appendix A contains optional material. The material provides mostof the details that were not shown in the body of the presentation.

Appendix B lists references.

Appendix A: Optional material

Optional material

Optional material (1/22)

The optional material slides provide most of the details that werehinted at in earlier slides but were intentionally omitted. SeeAppendix B if you want to see the complete details.

Since the material on these slides is optional, you will provide thedetails by working some exercises!

We assume that we are working with the hyperbolic paraboloidwhose equation is

u =v2

a2− w2

b2


Recall that when we looked at drawing the arcs of the upper andlower parabolas (see the slide whose header is Identify theparabolas (3/3)), we fixed a value of s and then stated that wewished to find the values of the degree 2 variables v and w so that

Upper parabola: u = v2

a2≤ s2

Lower parabola: u = −w2

b2≥ −s2

Exercise 1. Show that for the upper parabola v must satisfy−sa ≤ v ≤ sa.

Exercise 2. Find a similar inequality for the lower parabola andshow that your inequality is correct.


Recall that when we looked at drawing the arcs of the upper andlower hyperbolas (see the slides whose headers are Identify thehyperbolas (1/3) through Identify the hyperbolas (3/3)), wedid not give many details. The remaining slides and exercises fill inmost of the details.

The following table summarizes what we found for the hyperbolas.

Hyperbola Plane Hyperboloid

Upper hyperbola u = s2 1 = v2

(sa)2− w2

(sb)2

Lower hyperbola u = −s2 1 = w2

(sb)2− v2

(sa)2


We first look at the upper hyperbola.

Exercise 3. Find the equations for the upper branch and the lowerbranch of the upper hyperbola. (Hint: generalize what appears onthe slide whose header is Identify the hyperbolas (3/3).)(Answer: v = ± a

b

√(sb)2 + w2. Verify!)

Exercise 4. Restrict the domains of each of the branches of theupper hyperbola to the closed interval [−sb, sb ]. What are thecorresponding closed intervals for the ranges of each of thebranches? (The branches, restricted to the closed intervals, are thearcs.) (Hint: same hint as for Exercise 3.) (Answer for upperbranch:

[sa, sa

√2].)

Exercise 5. Verify that your answers to Exercise 3 and Exercise 4agree with the values shown for the example on the above-namedslide.


We next look at the lower hyperbola.

Exercise 6. Find the equations for the upper branch and the lowerbranch of the lower hyperbola.

Exercise 7. Restrict the domains of each of the branches of thelower hyperbola to the closed interval [−sa, sa ]. What are thecorresponding closed intervals for the ranges of each of thebranches?

Exercise 8. Verify that your answers to Exercise 6 and Exercise 7agree with the values shown for the example on the slide whoseheader is Identify the hyperbolas (3/3).


Note that we have not yet shown why the value s, as used for theparabolas and hyperbolas, guarantees that the four line segmentsdo, indeed, lie on the hyperbolic paraboloid. Nevertheless, graphsshown in the body of the presentation suggest that they are on thehyperbolic paraboloid.

Before we show why the line segments do lie on the hyperbolicparaboloid, we first view the equation of the upper hyperbola andthe equation of the lower hyperbola using parametric equations.

Recall that the hyperbolic cosine and the hyperbolic sine aredefined by the equations

cosh(t) = et+e−t

2 sinh(t) = et−e−t

2

and satisfycosh2(t)− sinh2(t) = 1


First we consider the upper hyperbola. Recall that it is theintersection of the plane u = s2 and the hyperboloid1 = v2

(sa)2− w2

(sb)2. The following table shows both the rectangular

equations and the parametric equations.

Rectangular Parametric

Upper branch u = s2 u(t) = s2

v = ab

√(sb)2 + w2 v(t) = sa cosh(t)

w(t) = sb sinh(t)

Lower branch u = s2 u(t) = s2

v = − ab

√(sb)2 + w2 v(t) = −sa cosh(t)

w(t) = sb sinh(t)

Exercise 9. Verify that the Parametric column is correct.


Recall that we restricted the domains of each branch of the upperhyperbola to the closed interval [−sb, sb ].

Let p = ln(√

2 + 1). Note that p > 0 since√

2 + 1 > 1.

Exercise 10. Verify that each of the following is true.

1 cosh(0) = 1 and sinh(0) = 0.

2 cosh(p) =√

2 = cosh(−p), sinh(p) = 1, and sinh(−p) = −1.


1 The endpoints of the upper arc of the upper hyperbola are(s2, sa

√2,−sb) and (s2, sa

√2, sb). The vertex has

coordinates (s2, sa, 0). (See Exercise 4.)

2 The parametric equations for the upper branch, restricted to−p ≤ t ≤ p, map one-to-one onto the upper arc.



1 The endpoints of the lower arc of the upper hyperbola are(s2,−sa

√2,−sb) and (s2,−sa

√2, sb). The vertex has

coordinates (s2,−sa, 0). (Use symmetry.)

2 The parametric equations for the lower branch, restricted to−p ≤ t ≤ p, map one-to-one onto the lower arc.

The table on the next slide summarizes what we’ve found.


Summary of results for upper hyperbola.


Upper branch u = s2 u(t) = s2

v = ab

√(sb)2 + w2 v(t) = sa cosh(t)

w(t) = sb sinh(t)Bounds for −sb ≤ w ≤ sb −p ≤ t ≤ p

upper arc sa ≤ v ≤ sa√

2

Lower branch u = s2 u(t) = s2

v = − ab

√(sb)2 + w2 v(t) = −sa cosh(t)

w(t) = sb sinh(t)

Bounds for −sb ≤ w ≤ sb −p ≤ t ≤ p

lower arc −sa√

2 ≤ v ≤ −sa


Exercise 13. Formulate exercises similar to Exercises 11 and 12 todescribe the lower hyperbola. That is, find the endpoints of thetwo hyperbolic arcs. Verify that p, as used in those exercises, isalso used for the parametric equations for the arcs of the lowerhyperbola.

The table on the next slide summarizes the results you should getwhen you formulate and solve Exercise 13.


Summary of results for lower hyperbola.


Upper branch u = −s2 u(t) = −s2w = b

a

√(sa)2 + v2 v(t) = sa sinh(t)

w(t) = sb cosh(t)

Bounds for −sa ≤ v ≤ sa −p ≤ t ≤ p

upper arc sb ≤ w ≤ sb√

2

Lower branch u = −s2 u(t) = −s2w = −b

a

√(sa)2 + v2 v(t) = sa sinh(t)

w(t) = −sb cosh(t)

Bounds for −sa ≤ v ≤ sa −p ≤ t ≤ p

lower arc −sb√

2 ≤ w ≤ −sb


We now have everything we need in order to see that the four linesegments lie on the surface of the hyperboloid.

The article An Interesting Property of Hyperbolic Paraboloidscovers a more general case than we’ve covered here. Two valuesof s are used, s1 and s2, for the planes u = s21 and u = −s22 . Twovalues of p, p1 and p2, are used to restrict the parametricequations for the arcs of the upper and lower hyperbolas.

The critical fact from the article is that the condition

s2 = s1e(p1−p2)

is both necessary and sufficient for the four line segments to lieprecisely on the surface of the hyperbolic paraboloid.

If we choose s1 = s2 then p1 = p2. This is precisely what we did!


We finish Appendix A by verifying that the points on the four linesdo, indeed, lie on the hyperbolic paraboloid. The steps shown hereare a modification of what is shown in the article An InterestingProperty of Hyperbolic Hyperboloids.

Note that we said lines, not line segments, in the precedingparagraph.

The next slide shows the graph of the hyperbolic paraboloidu = v2

a2− w2

b2, drawn according to the earlier optional material.

Look back at the slides Optional material (10/22) and Optionalmaterial (12/22) to see the bounds for all the arcs.

Note that the eight endpoints of the hyperbolic arcs have beenlabeled P1 through P8. Note the labels for the axes are u, v , andw .


P

P

P

P

P

P

P

P

1

2

3

4

5

6

7

8

w

u

v


The values for the coordinates of P1 and P5 are shown in thefollowing table.

Point Coordinates Point Coordinates

P1 ( s2, sa√

2, sb ) P5 (−s2, sa, sb√

2 )

P2 P6

P3 P7

P4 P8

Exercise 14. Fill in the coordinates for the remaining points.


Recall that if Q1 and Q2 are two specific points in 3-space, we canfind the parametric equations for the line passing through the twopoints. One method is described in the following 3 steps.

1 Let Q = (u, v ,w) be an arbitrary point in 3-space.

2 Q is on the line joining Q1 and Q2 if and only if the vectors−−→Q1Q and

−−−→Q1Q2 satisfy

−−→Q1Q = t

−−−→Q1Q2 for some real number t.

3 Set each of the three coordinates equal and solve for u, v , andw , yielding the desired parametric equations.

Let L15 be the line passing through P1 and P5. Let L26 be the linepassing through P2 and P6. Let L37 be the line passing through P3

and P7. Let L48 be the line passing through P4 and P8.

We next find the parametric equations of these four lines.


The parametric equations for L15 are shown in the following table.

Line Parametric

L15 u = s2(2t − 1)

v = sa[1 + t(

√2− 1)

]w = sb

[√2− t(

√2− 1)

]L26

L37

L48


Exercise 15. Verify that the parametric equations for L15 arecorrect. Find the parametric equations for the remaining lines.

We now will verify that all points on these four lines lie on thehyperbolic paraboloid.

We first verify that all points on L15 lie on the hyperbolicparaboloid. If (u, v ,w) is on the line, then there is a real number tsuch that

u = s2(2t − 1)

v = sa[1 + t(

√2− 1)

]w = sb

[√2− t(

√2− 1)

]In order to verify that (u, v ,w) lies on the hyperbolic paraboloid

we must show that u = v2

a2− w2

b2.



1 v2

a2= s2

[1 + 2t(

√2− 1) + t2(

√2− 1)2)

]2 w2

b2= s2

[2− 2

√2t(√

2− 1) + t2(√

2− 1)2]

We next show that v2

a2− w2

b2= u, which shows that the point does

lie on the hyperbolic paraboloid.


Exercise 17. Verify that the following computation is correct.

v2

a2− w2

b2= s2

[1 + 2t(

√2− 1) + t2(

√2− 1)2)

]−s2

[2− 2

√2t(√

2− 1) + t2(√

2− 1)2]

= s2[1 + 2t(

√2− 1) + t2(

√2− 1)2

−2 + 2√

2t(√

2− 1)− t2(√

2− 1)2]

= s2{−1 + 2t

[(√2− 1

)+√

2(√

2− 1)]}

= s2[−1 + 2t(

√2− 1)(

√2 + 1)

]= s2 [−1 + 2t(2− 1)]

= s2(2t − 1)

= u


Exercise 17 showed that all points on the line L15 lie on thehyperbolic paraboloid.

Exercise 18. Verify that all points on the other three lines also lieon the hyperbolic paraboloid. (Hint: for each of the lines, createand solve exercises similar to Exercise 16 and Exercise 17.)

Appendix A does describe all the steps needed to justify the bodyof the presentation. However, Appendix A lacks certain details andis short on motivation for some of the steps. Consult Appendix Bfor references to all the details.

Appendix B: References

There are two documents that contain additional details about theprocess described in this presentation. Both are located on the webpage

http://faculty.matcmadison.edu/alehnen/calculus2/Calculus 2 Home Fall 2011.html

Scroll down to the Selected Notes for Calculus II section.

1 John Ganci and Al Lehnen, How To Draw a HyperbolicParaboloid: Quick Guide, 2011.

2 Al Lehnen, An Interesting Property of Hyperbolic Paraboloids,2009.

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