Definition: The set of all points such that the sum of the distance from two points is a constant. (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet)
Algebraically:
Ellipses can be algebraically seen as the pictures above. An ellipse can be seen in two ways, "fat" and "skinny". How do we determine which is which? Well if the larger denominator, a^2, is under the x it will be a horizontal ellipse or "fat" ellipse. If the larger denominator, a^2, is under the x it will be a vertical ellipse or "skinny" ellipse. "A" give us the indication of the length of the major axis and the "b" gives us the indication of the length of the minor axis. With our "h" and "k" we can find our center. Do keep in mind that the "h" goes with "x" and "k" will always go with "y". With our equation we can find about everything needed to graph our ellipse. To find our a, b and c, we use "a^2-b^2=c^2".
By looking at the equation, we can find the "a" and "b". "A" pertains to our vertices, "b" to our co-vertices and "c" to our foci. Depending whether its horizontal or vertical, our vertices and foci have something in common, either the "x" or "y" for both vertices and foci will be alike. In finding our vertices if the bigger denominator(a^2) is under the y (vertical/"skinny" ellipse), our vertices and foci would have the similar x's. So what would we do with "a"? You add and subtract that to the "y" of the center two get your two vertices. This would be the same process for the "c" and foci. Finding our co-vertices would mean that the "y" would stay the same for both co-vertices. So in this case we would add and subtract "b" to our center to find our two co-vertices. This whole process would be the same for horizontal/"fat" ellipses except vice-versa.
By looking at the equation, we can find the "a" and "b". "A" pertains to our vertices, "b" to our co-vertices and "c" to our foci. Depending whether its horizontal or vertical, our vertices and foci have something in common, either the "x" or "y" for both vertices and foci will be alike. In finding our vertices if the bigger denominator(a^2) is under the y (vertical/"skinny" ellipse), our vertices and foci would have the similar x's. So what would we do with "a"? You add and subtract that to the "y" of the center two get your two vertices. This would be the same process for the "c" and foci. Finding our co-vertices would mean that the "y" would stay the same for both co-vertices. So in this case we would add and subtract "b" to our center to find our two co-vertices. This whole process would be the same for horizontal/"fat" ellipses except vice-versa.
Graphically:
Graphically, ellipses look like the picture above. The vertical/"skinny" ellipses consists of the same parts as the horizontal/"fat" ellipses. The center of our ellipse is found in our equation, as mentioned above. With our "h" and "k" we can find our center. Do keep in mind that the "h" goes with "x" and "k" will always go with "y". Our graph consists of a center, vertices, co-vertices, foci, a minor axis, a major axis and eccentricity. To determine the major axis and minor axis would only mean looking at the vertices and co-vertices. Looking at the vertices, whichever, x or y, is similar that would be the major axis. For the co-vertices, whichever, x or y, is similar that would be the minor axis. The only factor that has not been explained is eccentricity. Eccentricity is "c" divided by "a". This determines how much the ellipse deviates. If it's closer to one, it means that it looks more like a circle and vice versa if its closer to zero (0 < e < 1). Eccentricity is not necessarily seen on the graph but it helps the student to get an estimate of how the graph might look like when they draw it.
Here's more help with ellipses:
Real World Applications (RWA)-
Ellipses can be represented in the orbit of the planets in outer space. How do you say so? Looking at the picture demonstrates it perfectly and why. The sun would always serve as one of the foci. The other focus would be empty but still be there (http://oneminuteastronomer.com/8626/keplers-laws/). As described in the Kepler Law, "their orbits are similar ellipses with the commonbarycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus."(http://en.wikipedia.org/wiki/Ellipse#Ellipses_in_optimization_theory)
The things remaining around the ellipse have no physical "significance" but are still there. The basic real world of application of the ellipse is centered around the foci. One thing that is obvious from the picture above is the foci and the planet. It goes along with the concept of d1+d2= a constant. Which is what ellipses our all about! The picture above perfectly describes why an ellipse can be applied in the real world with the orbit of planets from outer-space.
URL's:
No comments:
Post a Comment