Parabolic
Reflectors
Gazing through a telescope, I see brilliant stars and other extraterrestrial spheres clearly. As I watch Monday Night Football, I wonder what allows the images to appear on the TV screen. Watching a large headlight beam light out into the night sky from a nearby airport, I try to figure out how they get all of the light beams focused into one long continuous stream of light. How does this all happen? Each one of the previous examples work because of something called parabolic reflectors. The parabola is used in so many different objects we use, and they are most often used because of their reflective properties. They are the reason that we see a continuous beam of light. It is because of parabolic mirrors that we are able to see images in the night sky magnified many times. Through its parabolic shape, satellite dishes have the ability to take in radio waves so that we can tune into a TV or radio signal. Parabolas, because of their properties of reflection, are used often in our life.
In
order to understand how a parabola is used in our lives, the parabola and its
key parts need to be discussed. A parabola can be defined as a locus of points
in a plane which are equidistant from a given point (the focus) and a given line (the directrix). The general equation of a
parabola is as follows: y=ax2+bx+c. Another name for the vertex form is
the quadratic equation. The standard
form of a parabola is as follows: y =a(b(x - h))2+k. One other general equation for parabolas is x2=4ay. The
graph of a parabola looks like this:
The vertex is the origin of a parabola, and is located at point (h,k) in a standard equation. The focus of a parabola is equidistant from any point P as the directrix is from point P. The directrix is a line that runs perpendicular to the axis of symmetry and is located opposite the side of the vertex from where the parabola opens up (for example, if the parabola opened up, the directrix would be under the vertex. The axis of symmetry in a parabola is as follows: y=h.
In the standard equation of a parabola, y =a(b(x - h)2)+k, “y” equals the locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix). The “a” value determines if the graph will be stretched vertically or not. In other words, if a=3, the parabola (each set of points) would be stretched vertically times x3. . If the “a” value is negative, the parabola will be flipped over the x-axis. The “b” value determines if the parabola will be stretched horizontally. However, if b=3, the set of points will not be stretched x3, but x1/3 (the reciprocal of b). If the “b” value is negative, the parabola will be flipped over the y-axis. The x value is the variable in the equation. The “h” value determines if there will be a horizontal shift of the parabola or not. For example, if h=1, there will be a shift right one. Note, however, that if h=1, and therefore a shift right of one, it will look like this in the equation: y =a(b(x - 1)2)+k. The “k” value determines whether or not there will be a vertical shift of the parabola. If k=2, for example, the graph will be shifted up 2 units. Here is an example of a standard equation of a parabola: y =2(3(x - 1)2)+4. The parabola (the set of points equidistant from the focus and directrix) is stretched vertically x2, is stretched horizontally x1/3, is shifted right 1, is shifted up 4, and is not flipped horizontally or vertically. The vertex is located at point (1,4), the axis of symmetry is y=1, and the graph opens up because it is not flipped over the x-axis. When the x and y values are switched, x=a(b(y - 1)2)+k, the parabola opens left and right, and is not a function. This is a basic review of the parabola and quadratic equation.
In order to understand the reflection properties of a parabola, the focus, directrix, and their
properties must be discussed more in-depth. The focus of a parabola in a standard equation, “f”,
is found by the following
equation: a= 1 . The directrix, a line perpendicular to the
4f
parabola’s axis of symmetry, is located outside of the parabola, (under it in standard position). The focus, (0,f) in a standard parabola, or (h, f+k) in a parabola that opens up and down, is at any time the same distance from any point “P” as a point on the directrix is when the directrix is perpendicular to the line formed by connecting “P” to the point on the directrix. Here is a picture that displays this:
http://en.wikipedia.org/wiki/Parabolas
When point “P” is the vertex, it follows that because the focus and
directrix are equidistant from point “P”, and the focus is at (0,f), the
equation of the directrix is y= -f.
Therefore, any point (x,y) on the parabola must be equidistant from
(0,f) and (x,-f). Applying this to real
life, the reflective property of parabolas comes from the fact that any point P
on a parabola is equidistant from the focus and directrix. When a light, sound, radio, or other type of
wave hits a parabola and is perpendicular to the directrix, it will bounce off
the parabola and hit the focus every time.
The picture on the next page shows this effect.
http://mathworld.wolfram.com/Parabola.html
The proof of the reflective property of parabolas uses a tangent line to
show how a wave or beam reflecting off of the parabola always will hit the
focus. When you have any point “A” on a
parabola and run a line through that point that is tangent to the parabola,
that tangent line will make equal angles with the line AF (the line from the
point to the focus) and with the line AA’ (the
line from the point on the parabola to the point on the directrix equidistant
from point A as point F). Here is a
picture that illustrates this proof:
http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml
One other unique aspect of the
reflective characteristics of parabolas is the fact that waves coming from the
under side of a parabola also reflect in a certain way. Waves that hit the under side of a parabola
that are perpendicular to the directrix will always bounce off the parabola at
the same angle (and will be a continuation of) the line formed by the distance
from the focus to the point on the parabola.
Therefore, when a wave beam hits the under side of a parabola, the waves
will bounce off in all directions except straight back.
In the real world, parabolas are used for reflective purposes in the form
of either a parabolic dish or a mirror such as ones used in telescopes.
Considering uses for parabolic reflectors, they may be used either for the
gathering or sending of energy such as light, sound, or radio waves because of
the reflective property discussed previously.
One of the uses of parabolas are as
light reflectors. For this use of
parabolas, a parabola-shaped mirror is used to reflect the light. In most uses of parabolic light reflectors,
there is a general light source nearby the parabolic mirror, and the light that
hits the parabola then is translated out (by the reflection property) into one
beam of light with parallel light rays.
One use of parabolic mirrors is in lights such as headlights and
spotlights. In most headlights, there
are either halogen or incandescent light bulbs which emit light in all
directions. In order to attain a focused
beam for better vision of the road at night, parabolic mirrors or reflectors
are placed behind the bulbs and are positioned at an angle best for the area
ahead of the car that needs to be seen.
For example, when you turn your “brights” on, the parabolic mirrors
behind those lights are positioned so that they focus the beams of light higher
up on the road for wider and longer visibility.
The same concept is used in searchlights, such as the ones utilized in
airports. In this case, there is a very
powerful light bulb behind a lens, and behind the bulb is a large parabola that
focuses the light from the bulb out into the night air. And in order that the beam may move, they
just put the search light on a swivel so that it can move back and forth but
still retain the focused beam. Another
example of how parabolic mirrors function in our lives is the telescope. In reflecting telescopes, there often is a
parabolic mirror placed at the end of the inside of the telescope. The parabolic mirror is made or positioned so
that the focus of the parabola is located at a second smaller mirror in the
front of the telescope placed at a 45 degree angle, and so when light hits the
parabolic mirror and is focused to the second mirror, that mirror then bounces
the image up through the eyepiece.
http://en.wikipedia.org/wiki/Reflecting_telescope
These are some of the ways that parabolas
mirrors are used for reflection of light.
Parabolas also reflect other types
of energy waves, such as sound and radio waves.
One of
the more modern ways that parabolas are used is in the case of parabolic
microphones. These are used for
eavesdropping, law enforcement purposes, and nature recordings and
observations. A parabolic dish is used
that has a microphone at the focus which picks up just about any sound up to
many meters away. The sound waves are
reflected off of the parabolic dish to the focus, from where they are
electronically transmitted or recorded so that the person utilizing the
parabolic microphone can then hear whatever hits the parabolic dish.
http://hyperphysics.phy-astr.gsu.edu/hbase/audio/mic3.html
As you can see, this could be very effective
in cases where a law enforcement officer or detective would want to eavesdrop
on a suspect, or when a scientist would want to listen to or record something
in nature without spooking anything or sitting in that spot all day and
night.
The
parabolic microphone is also used in audio recordings of sporting events. Another example of how parabolic dishes are
used is for the gathering of radio waves.
Satellite television utilizes parabolic dishes, or satellite dishes to
collect radio waves. A satellite dish is
positioned either on the ground, but more often on the roof, to collect radio
waves by focusing the radio waves coming from the satellite to the parabolic
dish into the focus, from where the signals are transported to a down converter
which then sends the signals into our TV’s as a picture. That and the use of parabolas as microphones
are other ways that our world uses parabolas as reflectors.
In our modern world, there are many
ways to use parabolas. However, without
first understanding them, a person wouldn’t even know what a significant role
parabolas play in our everyday life. As
one comes to understand parabolas, their focal properties, and how they allow
for reflection, one can appreciate and better understand how our modern world has
made use of parabolic reflectors in ways such as in the headlights, telescope,
parabolic microphones, and satellite dishes.
Bibliography
Nelson, R.
“Reflective Properties of Parabolas.”
www.cut-the-knot.org. 20
December 2006 <http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml>.
“Parabola.” Wikipedia.org. 20 December, 2006. Wikimedia Foundation, Inc. 20 December 2006 < http://en.wikipedia.org/wiki/Parabolas>.
“Parabolic Microphone.”
www. hyperphysics.phy-astr.gsu.edu. 11 January 2007.
<http://hyperphysics.phy-astr.gsu.edu/hbase/audio/mic3.html.>
“Parabolic
Microphone.” Wikipedia.org. 12 December, 2006. Wikimedia Foundation, Inc. 20 December 2006 < http://en.wikipedia.org/wiki/Parabolic_microphone>.
“Reflecting Telescope.”
Wikipedia.org. 15
December, 2006. Wikimedia Foundation,
Inc. 20 December 2006 < http://en.wikipedia.org/wiki/Reflecting_telescope>.
“Satellite Dishes.”
Wikipedia.org. 13
December, 2006. Wikimedia Foundation,
Inc. 20 December 2006 <http://en.wikipedia.org/wiki/Satellite_dish>.
Simmons, Bruce.
“Parabola.” www.mathwords.com. 14 April 2006.
11
January 2007. <http://www.mathwords.com/p/parabola.htm>.
Weisstein, Eric W. “Parabola.” WolframMathworld.com 3 August 2004. Wolfram Research, Inc. 20 December 2006 <http://mathworld.wolfram.com/Parabola.html>.
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