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Transcript of 010.AVI (Lecture 10)

By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

So please have a look at the screen here. And the yellow construction, which you see here is a
double cone.

... Prof. Nagaoka ...

Then you see a light green plane, which is intersecting the cone. And the lines of intersection
in this position are two hyperbolas. Now we can ...

... Prof. Nagaoka ...

And now we can rotate this plane and see all the various quadratic curves appear as intersection
lines. So here we have the parabola. And now we are changing more and more to an ellipse, and if
it is like this it is a circle. And we go again back to a hyperbola.

Next I want to show you how you can make in another kind of graphics, ... with polar coordinates
you can also describe the quadratic curves. Here you see two lines. This is the line which has
the point O ("oh") as a focus, and another line, which is called the directrix and it has the distance
d. Every point on the quadratic curve has the name P and the distance r from the focus. And between
the blue line and r we have the angle theta. And every quadratic curve is described by this
formula. That the distance from the focal point O is the socalled constant here, e for excentricity,
times d, which is this distance here, divided by one plus e (the same constant e, which we had here)
times the cosine of theta.

And depending on the value of the excentricity,

if it is between zero and one, we get an ellipse,

if it is just equal one, we have a parabola,

and if e is larger than one, we get hyperbolas.

So now I want to show you a computer graphics, which realizes this as well. Here we have
a value of the excentricity, which is between zero and one, so it is exactly an ellipse.

... Prof. Nagaoka ...

Now we increase it and just as it becomes one, we have the parabola. And if we continue to increase
the excentricity, we get another hyperbola half coming down from the top. And you see the two halves
of the hyperbola appear. So just by varying the excentricity, we can show all different quadratic
curves in one formula.

Thank you.

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Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).

Last Modified 16 January 2004

EMS Hitzer and Prof. R. Nagaoka are not responsible for the content of external internet sites.