RWA #1: Unit M Concept 5: Graphing Eclipses Given Equation

1. Mathematical Definition of an Eclipse: "the set of all points such that the sum of the distance from two points is a constant (Kirch)"

2.   The equation for an eclipse is . To define it graphically, an eclipse should look like an oval-shaped. There are two types of eclipses, which is either a "fat" or a "skinny" eclipse. In a "fat" eclipse,  the major axis has a length of 2a and the minor axis has a length of 2b. Moreover, the major axis lies according to the y-value since it has to stretch out horizontally. In a "skinny" eclipse, the major axis lies according to the x-value since it has to stretch out vertically. Both the "fat" and "skinny" eclipses' minor axis are opposite to its major axis. Graphically, an eclipse consists of the center (h,k), major axis, minor axis, foci points, vertices, and co-vertices. The center is the intersection of the major and minor axis. The vertices are endpoints to a major axis, while the co-vertices are endpoints to a minor axis. The foci are the focus points to demonstrate how much the eclipse deviates from being circular.
   To find the parts of an eclipse, you can determine by solving or putting the puzzle pieces together algebraically. In standard form, the x-value will go with h, and the y-value will go with k. The term a^2 and b^2 can either be under the term x and y depending what type of eclipse it is. The standard form of an eclipse will always be added and equaled to one. If it is not equal to one, make sure to simplify it. The standard form give you clues whether an eclipse is skinny or fat, center, major axis, minor axis, a, and b. Algebraically, the major axis will always be the bigger number and the bigger number will always be known as the term "a". Once you found your major axis, it is easy to find your minor axis by looking at the standard form. Thus, the term for minor axis is "b". To find the foci points or "c", you can use the equation a^2 - b^2 = c^2. Also, finding the eccentricity is very significant because it gives you an exact number of an eclipse that deviates from being circular. You can find the eccentricity by taking "c" and dividing it by "a" or c/a. The eccentricity of an eclipse is 0<e<1.


This image depicts the two types of ellipses and the key points both graphically and algebraically.

Here's a video about ellipses!!


3.  Ellipses can be found in our universe such as the solar system! The way the planets orbit around the sun has a shape of an ellipse including the satellites. Astronomers called this movement as elliptical orbits. For instance, the Earth's movement around the Sun has an eccentricity of 0.0167. In the solar system, the Sun is the focus. Since the Sun is the focus of an eclipse, the planets can either move closer or further away from the sun.
   Sometimes when you stare out of the sky, you can see the moon with a solar ellipse. This is "when the lunar disk passes directly between us and the sun." A lunar ellipse happens when the moon is considered a full moon. "Because the Moon casts a smaller shadow than Earth does, eclipses of the Sun tightly constrain where you can see them. If the Moon completely hides the Sun, even for a moment, the eclipse is considered total." The latest lunar ellipse was in December 2011 in Los Angeles. 

4. References:
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.mathopenref.com/images/coordgeneralellipse/Equation-3.jpg
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html
http://www.skyandtelescope.com/observing/highlights/237963491.html