Ellipse as an-example-learning- shifts-on-internet-era
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This document discusses the geometry of ellipses using Dandelin spheres. It presents equations to calculate the eccentricity and position of the directrix of an ellipse based on the radii of two imaginary spheres called Dandelin spheres. The document also mentions how this geometric model can be applied to other conic sections like hyperbolas and parabolas. It concludes by noting how the Internet provides helpful resources for fully describing ellipses and other conic sections using synthetic geometry.
![3
Eccentricity ratio
-If the point P is on ellipse and point
J is the projection of point P on line
t, and J' is the projection of P on line
t', (PJ=d, PJ'=d', d+d'=const).
-Triangles PJS and PTJ' are similar
(S is on the small circle and ST line
goes through V, PF=PS as tangents
to a sphere, JS || J'T), we have the
eccentricity equations:
PF/PJ=PS/PJ=PT/PJ'=PF'/PJ'. If
MN=2a, FF'=2c, and P is chosen to
be on either M or N point, d is now
an unknown, we have the equation:
(a-c)/d=(a+c)/(2a+d)----------(1)
From Equation (1) d=(a*a/c)-a and
e=(a-c)/[(a*a/c)-a)]=c/a, e<1.](https://image.slidesharecdn.com/ellipse-asan-example-internet-shift-170127035622/85/Ellipse-as-an-example-learning-shifts-on-internet-era-3-320.jpg)
![5
Position of the directrix as a function
of the radii of Dandelin's Spheres
Since a and c are known, Equations 2 and 3
provide b=SQRT(a*a-c*c)
In the slide #4 we found the directrix located
outside the ellipse at the distance [(a*a/c)-a]
symmetrically from M and N points (see slide
#3). Is convenient to refer to the distance from
the focus points, [(a*a/c)-a]+(a-c), therefore:
MJ=NJ'=(a*a-c*c)/c=
=(1/4)*[(r2+r1)*(r2+r1)-(r2-r1)*(r2-r1)]/c=
=2r1r2/SQRT[d*d-(r2+r1)*(r2+r1)]---------(5)](https://image.slidesharecdn.com/ellipse-asan-example-internet-shift-170127035622/85/Ellipse-as-an-example-learning-shifts-on-internet-era-5-320.jpg)
Ellipse as an-example-learning- shifts-on-internet-era
- 1. About ellipse, an example of learning shifts on Internet Danut Dragoi, PhD California 01/26/2017
- 3. 3 Eccentricity ratio -If the point P is on ellipse and point J is the projection of point P on line t, and J' is the projection of P on line t', (PJ=d, PJ'=d', d+d'=const). -Triangles PJS and PTJ' are similar (S is on the small circle and ST line goes through V, PF=PS as tangents to a sphere, JS || J'T), we have the eccentricity equations: PF/PJ=PS/PJ=PT/PJ'=PF'/PJ'. If MN=2a, FF'=2c, and P is chosen to be on either M or N point, d is now an unknown, we have the equation: (a-c)/d=(a+c)/(2a+d)----------(1) From Equation (1) d=(a*a/c)-a and e=(a-c)/[(a*a/c)-a)]=c/a, e<1.
- 4. 4 Eccentricity as a function of the radii of Dandelin's Spheres If we note with r1 and r2 , r2 >r1 , the radii of the Dandelin's spheres and d the distance between their centers (do not confuse this d with the variable PJ=d in slide #3) we can easily calculate FF' and MN as: FF'=SQRT(d2 -(r2 +r1 )2 )=2c--------------(2) MN=SQRT(d2 -(r2 -r1 )2 )=2a--------------(3) And the eccentricity e as: e=SQRT(d2 -(r2 +r1 )2 )/SQRT(d2 -(r2 -r1 )2 )-----(4)
- 5. 5 Position of the directrix as a function of the radii of Dandelin's Spheres Since a and c are known, Equations 2 and 3 provide b=SQRT(a*a-c*c) In the slide #4 we found the directrix located outside the ellipse at the distance [(a*a/c)-a] symmetrically from M and N points (see slide #3). Is convenient to refer to the distance from the focus points, [(a*a/c)-a]+(a-c), therefore: MJ=NJ'=(a*a-c*c)/c= =(1/4)*[(r2+r1)*(r2+r1)-(r2-r1)*(r2-r1)]/c= =2r1r2/SQRT[d*d-(r2+r1)*(r2+r1)]---------(5)
- 6. 6 Types of sections in a double cone that can be treated with the model described in here for ellipse Circle-------Ellipse-----Hyperbola--Parabola http://math2.org/math/algebra/conics.htm
- 7. 7 Conclusion Using the learning shifts suggested in a previous posting, link below, the Internet is found very helpful on fully describing the ellipse utilizing the synthetic geometry in space and plane. The method shown could be applied easily to other conics, hyperbola, and parabola. https://www.linkedin.com/pulse/learning-shifts-internet-era-danut-dragoi?trk=mp-reader- card