Conic Sections: Ellipse

1. What is the mathematical definition of this conic section and how does that definition play a role in the properties of the conic section and how it is shaped or formed?

 The mathematical definition of an ellipse is the sum of the distances from the 2 foci points to any point on the ellipse will ALWAYS be equal. This definition plays a role in the properties an an ellipse and how it is shaped because the distance from the foci to the center varies and correspondingly shapes an ellipse more "fat" or more "skinny". Moreover, an ellipse is formed by a slant cut into a double napped cone.

Check out this awesome website where you can interact and understand for yourself the definition of an ellipse: http://www.mathsisfun.com/definitions/ellipse.html  


2. How does the focus (or foci) affect the shape of the conic section?  
If the foci are close to the center, the ellipse will be closer to a circle. If the foci are farther from the center, the ellipse will look more like an oval. The amount an ellipse is "squished" away from being a circle is its eccentricity. The value of an ellipse's eccentricity is e= c/a. Since the foci are closer to the center than the vertices are, we know that c < a, and therefore, the value of e will always be less than 1. If the foci of an ellipse were brought in closer towards the center, it would be closer to being a circle. In fact, if the foci were brought in all the way to the center to let c=0, the eccentricity would also be zero when we divide 0/a and the ellipse would now be a circle. So, we can tell that the eccentricity will be 0 < e < 1. 
 
3. How do the properties of this conic section apply in real life? 

Tycho Brahe, a Danish of noble decent, was known for his accurate astronomical and planetary observations. Astronomy incorporates ellipses in the way that planets orbit in the solar system. They do not orbit in a circular fashion but rather in an elliptic fashion with the sun at one of the focus points. The Tycho Brahe Planetarium in Copenhagen, Denmark building represents an ellipse, as well, and the building was named after him.

Ellipses are also found in medicine. Lithotripsy is a method for destroying kidney stones. It uses shock waves to crush kidney stones into tiny pieces that are easier for the body to dispose of. The patient's kidney stones are aligned at one of the focus points and the shock waves at the other focus point destroy the kidney stones. This is a very useful and important application of the reflective properties of an ellipse from foci point to foci point.

"The Whisper Chamber" at the United States capital is another example of an ellipse applied in real life. It is said that John Quincy Adams figured out the phenomenon of the voices traveling across the room because of the elliptical shape and situated himself at the focus point on one end to eavesdrop on other members of the House.

Qualcomm Stadium at the San Diego Charger's home is another example of an ellipse application in real life. The major axis runs through the scoreboard while the minor axis runs perpendicular to it. This makes sense since a>b in ellipses. This conic section is used in most sport stadiums to allow the most  people possible around the rectangular field.




Works Cited

http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_ellipse.xml

http://cseligman.com/text/history/ellipses.htm

 http://www.mathsisfun.com/definitions/ellipse.html

https://sites.google.com/site/arinsellipse/real-life-examples 

https://sites.google.com/site/httphappyellispe78com/is-this-real-life
 
 Tycho Brahe Planetarium- courtesy of Google.