We observed earlier that the graph of
|
|
is the tangent plane to the
hyperboloid 
at the point
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This
plane intersects the 
plane ( 
) in the line 
in the 
plane,
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since 
.
If 
.
Now suppose that 
is an arbitrary point in the plane. Letting our unit vector
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it is clearly perpendicular
to the line, and there is a unique number 
and point 
such that
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The
number 
is the distance from 
to the line 
.
Now it is not difficult to see that if 
is the dihedral angle between the planes 
and the graph of 
(that is, the angle formed by intersecting these planes with a plane perpendicular
to 
),
then for each 
,
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Note
that 
. I will interpret the line 
as the directrix of the ellipse, and the number 
as the eccentricity in the relativistic context in the section Interpretation
of the Experiment.
For
now, I will simply determine the number 
in terms of 
. First of all, it is easy to see that the major axis of the ellipse is generated
by 
and that the intersection of 
and the line generated by 
is the point
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This tells us that the distance
from the directrix 
to the origin, a focus of our ellipse, is 
.
To
determine the number 
, it will be enough to calculate
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for a single point 
.
Since the origin 
,
we calculate
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and so we conclude that
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When
we interpret the ellipse 
dynamically using Special Relativity, we will be able to conclude that 
is the eccentricity of the ellipse that was characterized above.
.
,
,
.
.
.
, 
.