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Tuyen Nguyen Cluster 6: Mathematics

HYPERBOLIC PARABOLOID Abstract:

In Manchester, Greater Manchester (United Kingdom), there is a footbridge called

Hyperbolic Paraboloid Bridge, which was built in 1999. What is a hyperbolic paraboloid (HP)?

Basically, it is a saddle shape. For example, if you twist a square mesh, you will get a HP. This

shape can be created by paper folding using origami paper, as shown in the model I made. A

hyperbolic paraboloid is a double ruled surface. A surface is the boundary of a three-

dimensional figure. This means that a surface is the two-dimensional locus of points located in

three-dimensional space or a portion of space having length and breadth but no thickness

(dictionary.com). In this project, I will explain about the double ruled surface, the hyperbolic

paraboloid. The significance of the results is to explore what a HP is and how it changes after

connecting the vertices.

Figure 1: The origami model of the hyperbolic paraboloid

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Figure 3: Connecting vertices

Introduction:

The Hyperbolic paraboloid is a name that is not familiar to most people. However, there

are many mathematicians who study it. Theodore Olivier (1800s) was one of those who created

several beautiful models including the HP (see figure 1). Richard Rhodes recently made a

hyperbolic paraboloid sculpture, which was made of 500-year-old Chinese granite slabs. It was

shown during the construction in China in November 2002. The HP does not only relate in

Mathematics, but also is used for construction decoration. In 1964, the Venice Beach Pavilion

was built on the Gulf of Mexico. It was designed by architect Cyril T. Tucker and engineer

William Lindh. The pavilion’s roof was designed in the form of a hyperbolic paraboloid. Being

interested in the HP, my paper explores it as well as parabolas, hyperbolas and the hypothesis of

connecting the vertices.

Figure 2: The Olivier Model (left) and the Venice Beach Pavilion (right)

I want to know how I can find the hyperbolas, parabolas, and why it is a double ruled

surface. In addition, what do I construct if I connect the vertices A and C, and B and D together?

A C

B D

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In this project, I will show you some interesting things that I found out in my research of

the hyperbolic paraboloid.

Materials and Methods:

Most of the answers or directions on how to create a HP were Internet resources. I also

read books with visual graphics of different rotations of a three-dimensional shape (3-D shape) to

get more information and understanding of the HP, which is also three dimensional. I assume

that the reader knows the concepts of hyperbola and parabola. To get the answer for my third

question, I used tin foil to create a model and drew what I saw. I did this to get an idea of what

the shape would look like when I connected the vertices.

Discussion:

Double ruled surface:

A double ruled surface is that for every point on the surface, there are exactly two lines

that go through the point, and are contained in the surface. The only two double ruled surfaces

are the hyperbolic paraboloid and single-sheeted hyperboloid (1-sheeted hyperboloid).

Figure 4: Double Ruled Surface

Where are the lines in HP?

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Figure 5: HP, a double ruled surface

In the figure 5, we can see two lines l1 and l2 going through point A. They are also

contained in the surface above, which also represents the hyperbolic paraboloid.

Parabola:

It is not hard to find the parabolas in the HP because all you need to do is to look at it

horizontally.

What happens if I twist it a little more?

Figure 7: Uniplanar

A

B

l4 l5

l4 l5

(Wolfgang Boehm and Hartmut Prautzsch)

Figure 6: Parabola in HP.

l1

l2

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Looking at the curve above (from l4 to l5) we can see that it is a parabola. However, the

whole curve is uniplanar, which means it is situated or occurring in one plane. Since, the result

turns out not to be a parabola, the question is: Are there any special angles that we should use to

create a hyperbolic paraboloid?

Hyperbola:

The hyperbolas are more complicated to see because you will not see them easily unless

you cut the HP horizontally.

Figure 8: Cutting the HP

(The arrow in figure 8 shows the viewer’s direction.)

Figure 9: Hyperbolas in HP

If you notice the arrows, you will see the hyperbola has changed. That means that the

hyperbolas will change the direction (from horizontal to vertical) if we keep cutting downward.

cutting

cutting

1

2

1

2

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Figure 10: Transferring direction of hyperbolas

So, what happens in between them?

Figure 10

The equation for the horizontal hyperbola is:

x2 – y2 = 1 a2 b2 dsf

And the vertical hyperbola is:

y2 – x2 = 1 a2 b2 dsf

First, let a be the length of the major axis and n is any positive number. The a-value will

decrease from n to 0. Since, the a-value gets closer to zero, it is not a hyperbola because the

asymptotes will cross the other branch. After that, a-value begins increasing from 0 to n and the

shape (the graph) will transfer to be the vertical hyperbola.

Horizontal hyperbola

Vertical hyperbola

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Connecting:

Connecting the vertices A and C, B and D together will make the shape change.

Figure 11

We cannot make it out of paper because it requires a flexible material to create the shape.

And the result turns out not to be a hyperbolic paraboloid.

Let us call it the 8-shape. This 8-shape has two parts and both of them have the same

minor axes, which is the distance between the center and the projection of A onto x-axis. They

also have the same major axes, which is the distance from the center to the vertices. This 8-

shape is also three-dimensional.

Figure 12

However, it is not a ruled surface, which is a surface such that for every point of the

surface, there is a straight line passing through the point, contained in the surface. The 8-shape is

made of curves that begin at the same vertex and then return there after “traveling around”.

B

A

D

C

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These curves are the nodes. Finally, the 8-shape has many similarities with the hyperbola such

as the minor and major axes, center and vertices but it is not two-dimensional. I am still working

on finding the result or the equation for the graph. (figure 11)

Bibliography:

• Boehm, Wolfgang and Prautzsch, Hartmut. Geometric Concepts For Geometric Design:

Algebraic Surfaces (305-307)

• <http://www.amherst.edu/~amcastro/Mathmedia/Galleries/Surfaces/Hiplar.html>

• Weisstein, Eric W. Wolfram Research, Inc.

• <http://www.ualberta.ca/dept/math/gauss/fcm/calculus/multvrb/grph_hyprblc_prbld.htm>

• Sharp, John.

<http://www.mathsyear2000.org.explorer/slice/surface.shtml>

• http://www.thok.dlc/hyperbol.html

• Demaine, Erik.

<http://theory.lcs.mit.edu/~edemaine/hypar>

• http://www.structurae.de/en/structures/data/str02093.php

• http://www.ceciliacotton.ca/archives/00000188.htm

• http://dictionary.reference.com/search?q=surface