applications of ellipses & Hyperbolas
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Transcript of applications of ellipses & Hyperbolas
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.