Monday, February 10, 2014

RWA #1: Unit M Concepts 4-6: Conic Sections in real life.

Ellipses (Concept 5)

1. The mathematical definition of an ellipse, according to Mrs. Kirch is, "the set of all points such that the sum of the distance from two points in a constant."


2. Key features of an ellipse include the center, major and minor axis, two vertexes, two foci, two co-vertices, and the ellipse's eccentricity. 

This conic section algebraically looks like:  





See full image here 

The ellipse is cut by a slant through a cone, it cuts a shape of an elongated circle, an oval. Here are some photos and video for you reference. 


See first image here (graph)
See second image here (cone)

GRAPHICALLY & KEY FEATURES

Graphically you can easily see if your ellipse will be skinny or fat visually. If it is skinny, you will know your major axis will be x = and if it is fat, you will know your major axis will be y =. You can determine your center by finding the point at which the major and minor axis intersect and it will be defined as (h,k). H always goes with x and k always goes with y. The vertices, co-vertices, and foci will be easy to identify on a graph. Remember that the vertices, foci, and major axis will all have a certain number in common.


CENTER AND MORE

You can find multiple key features from the standard form of the equation alone. The center will always be (h,k) and h will always be with x and k will always be with y. 

AXIS - FAT OR SKINNY?

 The bigger denominator will be a^2 and the smaller denomiator will be b^2. The bigger denominator will determine if your ellipse will be horizontal "fat" or vertical "skinny." You only need to see if the bigger denominator is under the x^2 term, then it will be horizontal. If it is under the y^2 term it is going to be vertical. A fun little trick to remember if it will be vertical is "(wh)Y you so skinny?The major axis is the factor in making the ellipse horizontal or fat since it is the longest diameter, the minor axis is the shortest by default. If the ellipse is horizontal the major axis will be y = and if it is vertical the major axis will be x = [another neat trick, "(wh)Y you so X-tra skinny?" since if it is vertical and skinny, the major axis will be x.] 

VERTICES GALORE

The vertices are expressed as 2 points that lie on the major axis.Co-vertices are the points at the end of the minor axis. The bigger denominator's (a) square root will determine how many units away from the center you will plot the vertices and the distance between the two vertices (the major axis, usually drawn with a solid line graphically) will equal 2a. Plot a units to the left and right of the center if x^2  has the bigger value under it. Plot  a units up or down from the center if y^2 has the bigger number under it. The smaller denominator's (b) square root will also determine how far away you will plot the co-vertices from the center and the distance between the two co-vertices (the minor axis, which is perpendicular to the major axis graphically. Also typically drawn dotted) will equal 2b. Plot the co-vertices b units up and down from the center if x^2 has the bigger value under it or left and right from the center if the y^2 value has the bigger number under it.  The foci can be found with the equation: a^2 - b^2 = c^2. C (foci) is also associated with the major axis and vertices, and will have some of the same coordinates in common. The closer the foci is to the center, the more circular the ellipse will be. The farther away the foci is from the center , the more it deviates from a circle, meaning its more stretched out. Now that we have "c" we can find the ellipse's eccentricity with the equation "c/a" and an ellipses's eccentricity is 0 < e < 1. The closer an the eccentricity is to 1, the skinnier the ellipse will be, the farther it is from 1 (closer to 0) the fatter the ellipse will be.  

For more clarification please reference to this video giving you a quick introduction to ellipses. 




3. A real world application of an ellipse can be seen in architecture through "whispering galleries." One of the properties that an ellipse has is a reflective one. "If a tangent line were drawn at point P, the angles formed by that line and the lines PF and PF would be equal. Thus a light or sound wave emanating from one focus of an elliptic surface would be reflected to the other focus." (http://teachertech.rice.edu/Participants/dchipman/lessons/trajectories/ellipse.htm) //associated with the last image. A most famous example of such is in St. Paul's Cathedral in London. 




 Based on these two images, we can see that at the foci of the ellipse, sound is reflected from one focus to another through a direct line (the major axis.) But an essential key to all of this is the shape of course. The ellipse's curves are able to reflect soundwaves that hit it and bounces it off the walls, focusing them to several locations in the room. However if you were to stand at the very endpoints of the ellipse, the range of the soundwaves would not be as far or wide and would not reflect as well as the foci. And should you stand at the co-vertices, the results would be less desirable since the co-vertices fall on the minor axis and would not provide a sufficient range for the soundwaves to reflect off of. So if you are an eavesdropper and you're standing in the right place, you can hear a slightly audible conversation happening maybe 30-40 feet away.


4. Works Cited
  • http://www.mathopenref.com/ellipse.html
  • http://mathforum.org/sanders/geometry/GP18Ellipse.html
  • http://www.stpauls.co.uk/
  • http://www.msichicago.org/whats-here/exhibits/whispering-gallery/
  • http://www.youtube.com/watch?v=5nxT6LQhXLM

1 comment:

  1. Awesome post. Keep doing what you do. Enjoy Maths

    ReplyDelete