Hyper BoloiId Model

2INDEXPART 1: Theory of the Hyperboloid Page 3

PART 2: Applications of the Hyperboloid Page 9

PART 3: Construction of the Model Page 13

4CONICOIDSA quadric surface is a called a conicoid because any plane sectionof a quadric surface produces a conic section (parabola, ellipse, andhyperbola) or a limiting case of one (lines or a point).

5The Two Hyperboloids

Elliptical Hyperboloid

The hyperboloid of 1 sheet, as shown above in figure 3, sometimesdescribed as an elliptical hyperboloid, has ellipses for plane sectionsparallel to the xy-plane.

Plane sections parallel to the z-axis are hyperbolas.

6

figure 3

Hyperboloid Traces

The three equations for the traces in the principal planes of thehyperboloid of one sheet are shown above

7

figure 3

Hyperboloid of Revolution

When a = b in equation (1), the elliptical hyperboloid of onesheet becomes the hyperbola of revolution and the tracesparallel to the xy-plane become circles. In the model for part C,a = b = 4 and c = 5. By substituting these values into (1), we get (5), the equation for the surface for the model in part C.

8

figure 3

9Applications of the Hyperboloid

There are simple applicationsfor the hyperboloid in the formof metal or natural !bers.

Another simple application isthe use of wooden laths to buildcircular-form trellises for the garden.

In all of these applications, the hyperboloid is one of revolutionand the structure is a ruled surface. The entire structure canbe formed with straight line-segment elements and circularcross-sectional parts for support.

10

Another Application

The nuclear cooling towers are semi-hyperboloids. They approximate thehyperboloid of revolution and help exhaust hot air from the reactor.

11Application in Spain

The beautiful Gothic cathedral in Barcelona, Spain has large hyperboloidsupported by 4-large central pillars, surrounded by 12 smaller hyperboloidstructures. The cathedral was started in 1882 and the famous architectAntoni Gaudi (1852-1926) developed its basic design.

12

Application in Japan

Kobe Port Tower is a hyperboloid structure, 108 m high. It is a lattice tower in the port cityof Kobe, Japan. Observe the red-steel beamsemphasing that it is a ruled surface.

It has an observation deck which o!ersspectacular views of the bay and thesurrounding area.

The tower was completed in 1963.

HyperboloidModel

The theory of the hyperboloid was developed in part A. It was shownthat the hyperboloid of revolution is a special case. You will useequation (5) from part A as the basis for constructing your model.

13

Basic Tools You Needfor the Hyperboloid Model

14

Compass

Protractor

Ruler (with centimeter scale)

Scissors

Basic Materials You Needfor the Hyperboloid Model

15

Paper for Diagram:any plane paper or white posterboard,about 14 in x 19 in(40 cm x 50 cm)

Posterboard for Parts:(4-ply or 6-ply)

1 sheets poster board in one color,color on both sides, about 22 in x 28 in(55 cm x 70 cm)

1 sheets poster board in contrasting color,color on both sides, about 22 in x 28 in(55 cm x 70 cm)

16

You will construct a template similar to the one shown in !gure 4. The detailedconstruction is shown on the pages whichfollow.

From the diagram of !gure 4, you will takemeasurements to construct 14 circulardisks which interlock with slots to formthe model.

BASIC DIAGRAM FOR THE PARTS

FOR YOUR MODEL

!gure 4

17

!gure 5

The 14 slotted disks shown represent the parts of your model. The parts areconstructed using the basic diagram of !gure 5.

PARTS FOR THE HYPERBOLOID MODEL

18

Figure 6

BEGIN THE BASIC DIAGRAM

50 cm

40 cm

Draw two perpendicular lines, centered on your paper. Use a compassor a protractor for accuracy.

O

19

Figure 7

50 cm

40 cm

O

SCALE THE AXES

Mark 6 equal spaces of 6 cm along the horizontal axis. Then, mark6 equal spaces of 4 cm on the vertical axis, as shown in !gure 7.

4 cm

6 cm

20

Figure 8

O

ADD PARALLEL LINES

Pass the 6 parallel lines through the marked spaces as shown in !gure 8.

21

Figure 9

O

ADD MORE PARALLEL LINES

Pass the 6 more parallel lines through the marked spaces as shown in !gure 9.

22

Figure 10

O


Pass the 6 more parallel lines through the marked spaces as shown in !gure 9.

23

Figure 10

O


Use a ruler or compass to !nd the midpoint of AB at M and themidpoint of BC at N as shown in !gure 10.

24

Figure 11

FINISH THE PARALLEL LINES

Pass a line through the points O and M. Then, pass a line throughthe points O and N as shown in !gure 11.

O

25

Figure 12

REMOVE SOME LETTERS

After you complete all of the parallel lines in your diagram, remove the lettersA, B, C, M, N and leave O, as shown in !gure 12. This is a necessary to step so thatthe same letters can be used in the steps to follow, but for di!erent points.

O

26

Figure 13

Label AXES

Label the vertical axis with variable z and the horizontal axis with variable x.You will use a centieter ruler to plot data for a hyperbola curve, shown on thenext page, page 27, in !gure 14.

O

27

TEMPLATE REQUIREDYou will calculate data and makea hyperbola graph on the gridyou made in !gure 13. Whenyou !nish you will have thediagram similar to the one in!gure 14.

!gure 14.

28

EQUATION FOR THE MODEL

Equation (5) represents is the mathematical equation representingthe surface of the model you build. It was developed on page 13 ofpart A, where the theory of hyperboloids was developed.

When y = 0 in (5), it simpli!es to the xz-trace, the equation of thegraphy you will need for a template. The equation is designated asequation (1) above.

1

When y = 0,

29

SOLVE THE EQUATION

Solve equation (1) above for z. The solution is shown on the next page.

1

30

SOLUTION

Equation (1) is solved in the steps above. You use equation (9) to computethe data needed for the hyperbola template shown in !gure 14, on page 27.

31

DATA FOR GRAPH

Equation (9) is the equation for hyperbola template shown in !gure 14, on page 27.You need to compute values for (x,z) and plot the graph on your grid.

( 5

25 9 (3) 7.5

(x,z) = (( 5,( 7.5) or (x,z) D {( 5, 7.5),(5, -7.5),(-5, 7.5),(-5,-7.5)}

32

CALCULATE YOUR DATA

Complete the Table 1. The valuesfor x = ( 5 were computed in thesample on page 31.

Use a calculator to approximateany of the square roots to thenearest tenth.

Check your answers on page 33.

(cm) (cm)

TABLE 1

33

CHECK YOUR DATA

Table 1 is shown completedfor you to check your answers.

(cm) (cm)

TABLE 1

34

Hyperbola Graph

Plot the data youcalculated in Table Ion your basic grid.

Use a centimeter rulerto locate the points onthe graph.

The (x,z) = (6,11.2) isshown measured fromthe axes.

(4,0)

(5,7.5)

(6,11.2)

(7,14.4)

(8,17.3)

!gure 15

35

Label PointsIn !gure 16, the segmentsAB, CD, EF, and GH haveadditional points labeledfrom !gure 15.

Label the interesectingpoints on your diagramto match those shown:I, O, J on segment AB;K, L, M on segment CD;N, P, Q on segment EF:and R and S on segment GH.

(4,0)

(5,7.5)

(6,11.2)

(7,14.4)

(8,17.3)

!gure 16

36

MEASUREUse a centimeter rulerto measure the lengthsof the segments: AB, CD,EF, and GH. Write thelengths of the segmentsin Table 2.

(4,0)

(5,7.5)

(6,11.2)

(7,14.4)

(8,17.3)

!gure 17

Length (cm)

TABLE 2

37

Divide the Lengths

From Table 2, take half of the lengths of eachsegment you measured and write the number in the spaces provided in Table 3.

Length (cm) Length (cm)

TABLE 3 (4,0)

(5,7.5)

(6,11.2)

(7,14.4)

(8,17.3)

!gure 18

38

Constructing Parts 1 and 2

Use the radius for AB from Table 3 to construct the two circular disks shownin !gure 19. Draw a diameter on the disk and label the end points A and B.Then use your basic grid of !gure 17 to locate the points I, O, and J by measuringthe spaces from the grid and marking them on the diameter of AB.

Part 1 Part 2

!gure 19

39

Making the Slots for Parts 1 and 2

Use a protractor or compass to construct perpendicular lines at points I, O, and J.Then, cut very thin slots using a scissors. You need to cut twice to make the smallspace on each slot, as shown in !gure 20.

Part 1 Part 1

Part 2 Part 2!gure 20

40

Constructing Parts 3, 4, 5, and 6

Use the radius for CD from Table 3 to construct the four circular disks shownin !gure 21. Draw a diameter on the disk and label the end points C and D.Then use your basic grid of !gure 17 to locate the points K, L, and M by measuringthe spaces from the grid and marking them on the diameter of CD.

!gure 21

Part 3

Part 5 Part 6

Part 4

41Making the Slots for Parts 3, 4, 5, and 6

Use a protractor or compass to construct perpendicular lines at points K, l, and M.Then, cut very thin slots using a scissors. You need to cut twice to make the smallspace on each slot, as shown in !gure 21.

!gure 21

Part 3 Part 3

Part 5

Part 4

Part 6 Part 5

Part 4

Part 6

42

Constructing Parts 7, 8, 9, and 10

Use the radius for EF from Table 3 to construct the four circular disks shownin !gure 22. Draw a diameter on the disk and label the end points E and F.Then use your basic grid of !gure 17 to locate the points N, P, and Q by measuringthe spaces from the grid and marking them on the diameter of EF.

!gure 22

Part 7

Part 9

Part 8

Part 10

43

Making the Slots for Parts 7, 8, 9, and 10

Use a protractor or compass to construct perpendicular lines at points N, P, and Q.Then, cut very thin slots using a scissors. You need to cut twice to make the smallspace on each slot, as shown in !gure 23.

!gure 23

Part 7 Part 8

Part 9

Part 7

Part 9 Part 10

Part 8

Part 10

44Constructing Parts 11, 12, 13, 14

Use the radius for GH from Table 3 to construct the four circular disks shownin !gure 24. Draw a diameter on the disk and label the end points E and F.Then use your basic grid of !gure 17 to locate the points R and S by measuringthe spaces from the grid and marking them on the diameter of GH.

!gure 24

Part 11 Part 12

Part 11 Part 12

45

Making the Slots for Parts 11, 12, 13, and 14

Use a protractor or compass to construct perpendicular lines at points R and S.Then, cut very thin slots using a scissors. You need to cut twice to make the smallspace on each slot, as shown in !gure 25.

!gure 25

Part 11 Part 11

Part 13 Part 13

Part 12

Part 14

Part 12

Part 14

46

ASSEMBLE THE DISKS

Figure 17 is very important in assembling the 14 slotted disks of the model.The disks slide together by positioning the slots as shown in !gure 26. Startfrom the center with the two smallest disks, part 1 and part 2. The orange disksare parallel to each other and the blue disks are paralllel to each other.

!gure 26

Part 1

Part 2

47

COMPLETE THE ASSEMBLY

When you position all of the disks in their slotted positions, they match the diagramof !gure 17 and you have completed the model. This is illustrated by !gure 27.

!gure 27

Documents

Hyper BoloiId Model