Monday, February 10, 2014

RWA #1: Unit M Concept 5: Ellipses

1). An ellipse is the set of all points such that the sum of the distance from two points known as the foci is a constant. (http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet)

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php
2). Representing an Ellipse Graphically: An ellipse consists of a central point known as the center, 2 vertices connected with the major axis, and 2 co-vertices connected with the minor axis. The distance from the vertex to the center is known as 'a' and the distance from the co-vertex to the center is known as 'b'.




https://people.richland.edu/james/lecture/m116/conics/conics.html

Ellipses also have 2 points known as the foci. As demonstrated in the picture to the left, The sum of the distance from these two points (d1 + d2) is always a constant (2a). This is essentially what makes an ellipse. Also, the distance from the focus to the center is known as c..



http://coachjpocconics.weebly.com/ellipses.html
Algebraically:  As the picture suggests, there are two different ellipses with their corresponding equations which differ slightly with the placement of 'a' and 'b'. If 'a' (the bigger value) is under the y term, then the ellipse will be a vertical/ skinny ellipse and if it is under the x term, then the ellipse will be horizontal/ fat. 'H' and 'k' represent the x and y coordinates of the center.

Important: It is significant to know the eccentricity regarding an ellipse. Eccentricity is how round a conic is, 0 being perfectly round. An ellipse's eccentricity is greater than  0 but less than 1. This could be written as  e = 0 < x < 1. This is because an ellipse isn't perfectly round like a circle (whose eccentricity is 0) and yet it is more round than a parabola (whose eccentricity is 1). Eccentricity could be found by dividing 'c' by 'a' (c/a).

3). Real World Application: Elliptical Training Machinery
Elliptical gym equipment, such as elliptical runners/climbers are very useful when trying to avoid joint injury in a workout. Basically, the elliptical motion which is usually amplified with the use of handle bars or a motor drive, limits the impact on the joints that is very common with running. This article outlines this use for ellipses as well as a few other uses: http://www.ehow.com/info_8522010_real-life-uses-ellipses.html
...And this is a guy with a cool accent who will transmit elliptical knowledge to your brain in a brief 2 minutes:

4). URLs cited : Pictures from the top-down:
http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

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