Soran University

Faculty of Engineering

Department of Chemical Engineering

Name: Abdulsamad Alhamawande

Title: applications of ellipses & Hyperbolas.

Supervised by: Fouad Naderi

Date: 2015/04/17

Applications of Ellipses:

1. Football

If an ellipse is rotated about the major axis, you obtain a

football.

2. Satellite and Planet Orbits

Kepler's first law of planetary motion is:

The path of each planet is an ellipse with the sun at one

focus.

3. Whispering Galleries -- in the old House of

representatives

Statuary Hall in the U.S. Capital building is elliptic. It was in

this room that John Quincy Adams, while a member of the

House of Representatives, discovered this acoustical

phenomenon. He situated his desk at a focal point of the

elliptical ceiling, easily eavesdropping on the private

conversations of other House members located near the

other focal point.

4. Whispering Galleries -- Mormon tabernacle

The Mormon Tabernacle in Salt Lake City has an elliptical

ceiling. You can hear a pin drop from 175 feet away.The

Tabernacle is 250 feet long, 150 feet wide, and 80 feet high.

The organ has 11,623 pipes!

5. Elliptical Pool Table

The reflection property of the ellipse is useful in elliptical

pool --if you hit the ball so that it goes through one focus, it

will reflect off the ellipse and go into the hole which is

located at the other focus.

6. The Ellipse in D.C.

The Ellipse near the White House in Washington, DC is aptly

named.

Applications of Hyperbolas: 1. Dulles Airport

Dulles Airport, designed by Eero Saarinen, is in the shape of

a hyperbolic paraboloid. The hyperbolic paraboloid is a three

dimensional curve that is a hyperbola in one cross-section,

and a parabola in another cross section.

2. Lampshade

A household lamp casts hyperbolic shadows on a wall.

3. Gear transmission:

Two hyperboloids of revolution can provide gear transmission

between two skew axes. The cogs of each gear are a set of

generating straight lines.

4. Sonic Boom

In 1953, a pilot flew over an Air Force Base flying

faster than the speed of sound. He damaged every building

on the base. As the plane moves faster than the speed of

sound, you get a cone-like wave. Where the cone intersects

the ground, it is an hyperbola. The sonic boom hits every

point on that curve at the same time. No sound is heard

outside the curve. The hyperbola is known as the "Sonic

Boom Curve." In the picture below, the sonic boom is "visible"

due to the humidity. The photo below was taken by Ensign

John Gay, U.S. Navy from the

aircraft carrier Constellation. Sports

Illustrated and Life both ran the photo.

5. Cooling Towers of Nuclear Reactors

The hyperboloid is the design standard for all nuclear cooling

towers. It is structurally sound and can be built with

straight steel beams.

When designing these cooling towers, engineers are faced

with two problems:

(1) the structutre must be able to withstand high winds and

(2) they should be built with as little material as possible.

The hyperbolic form solves both of these problems. For a

given diameter and height of a tower and a given strength,

this shape requires less material than any other form. A

500 foot tower can be made of a reinforced concrete

shell only six or eight inches wide. See the pictures below

(this nuclear power plant is located in Indiana).

Graphing a Rotated Conic

If you are asked to graph a rotated conic in the

form , it is first necessary to

transform it to an equation for an identical, non-rotated conic. This is then plotted onto

new axes which are drawn onto the graph. The equation for the nonrotated conic can be

found by: . Note how capital

letters are used for the pronumerals. This signifies that they reperesent different values to

the original equaiton.

To determine the values of X and Y, you use the formulae:

By substituting the new values of and into the original equation, a new one can be

obtained which represents a non-rotated conic which can be plotted on a set of axes

rotated at (anti-clockwise) to the original x- and y- axes. When you do this, however, it

will still be necessary to determine the new rotated location of points such as the vertex,

foci and directrixes. This can be done using the following formulae:

Where (X,Y) are the new rotated coordinates of the orginal point (x,y).

The formula: can be used to determine the type of conic from the original

equation before you start graphing:

: Parabola

: Ellipse

: Hyperbola

Rotating a Conic

If you wish to rotate a conic by a certain angle, , it is relatively simple. All you do is

make the following substitution from the previous section:

Replacing the and values from the function with these new ones. Then simplify the

answer, and it will be a function for the same conic rotated by counter-clockwise about

the origin.