Planes Intersecting Cones
On
this page, I develop some of the details of the plane-slicing-cone picture discussed
in the introduction. The relationship between the plane-slicing-cone picture
and the focus-locus, and focus-directrix properties will be developed in three
stages. On this page, I will sketch the strategy that I will use in 3-dimensional
Euclidean Geometry in a special case.
Recall that we use a "standard" cone in Euclidean
space given by the equation
|
|
as depicted below. In later sections, I will interpret such a picture as a "light cone" at the origin event.
The 
axes are of course, orthogonal
in the Euclidean metric.
Next,
consider the hyperboloid defined by the equation
|
|
I
said that each point 
in the interior of the cone determines a plane. Our special case will consist
in choosing to belong
to the upper half of the hyperboloid defined above. In special relativity, this
hyperboloid will have special significance. Select the point on the hyperboloid:
= 
for A and B arbitrary.
Let 
and assume that 
The
plane tangent to the hyperboloid at that point intersects the cone in an ellipse.
For
example, the center of this ellipse is 
. It has semi-major axis of length
generated by 
at 
, and it has semi-minor axis of length 1, generated by 
at 
.
From
this, it is easy to calculate that the foci are at points:
|
|
Now
this calculation tells us nothing about the focus-locus or the focus-directrix
properties of 
.
In fact, there is no directrix evident in the construction. For these, I consider
a different ellipse.
Consider
the plane 
called the 
plane that is perpendicular to the axis of the cone, and passes through
the vertex. And let 
be the Euclidean orthogonal projection to this plane. Then the ellipse that
I consider will be the projection of
to the
plane. I will denote that projection 
.
Let us first show that 
is an ellipse.
I
do this in two ways. First, I represent the curve implicitly, and then I use
an explicit parametric representation to cast it, after translation and rotation,
in the form (for some
)
Notice
that there is a natural choice for the directrix for this ellipse, which I will
justify below. Consider the intersection of the tangent plane, the graph of
a function of x and y:
|
|
with the
plane: .
Since we assume that ,
this intersection is the line:
Special
relativity will establish that
has the focus-locus property, and that the line
will be the directrix for
for which the focus-directrix property is true. We will see that on the Interpretation
of the Experiment page.
Characterization of similarity classes of ellipses:
We
are now in a position to describe the similarity classes of ellipses (that are
not circles) in 
plane. These classes are overspecified by requiring that they have:
)It
is enough in fact to choose the center
and then to
Select the point:
=
on the hyperboloid 
.
The plane tangent to the hyperboloid at that point intersects the cone in the
ellipse .
And then the projection
will satisfy the three conditions above. We will then also know the directrix
and the eccentricity of
following a calculation that I will make below.
If
we let the unit vector: 
,
so that 
we can picture the sequence of 3 points:
|
|
as the points in the 
plane occupied by a uniformly moving object with (classical or prerelativistic)
velocity 
at time 0, 1, and 2. When this picture is interpreted relativistically, we will
also have an interpretation of the ellipse 
itself as well as the focus-locus and focus-directrix properties.
In the picture below we specified only the point ,
the tip of the red arrow, to determine as above, the light blue ellipse and
its directrix and foci.
Now
let us interpret the directrix and eccentricity in this Euclidean setting. Later
I will give the relativistic interpretation.
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. And as we mentioned
above, special relativity will establish that
has the focus-locus property, and that the line
will be the directrix for
for which the focus-directrix property is true.
The
bridge between this Apollonian plane-slicing cone picture and the focus-locus
picture will be the following. Every ellipse that we construct via the 
process belongs to some hyperboloid
The planes that
are tangent to the unit hyperboloid
as in the calculation above
form a special class.
The intersection of the latter with the cone will be called "boost intersections."
The
boost intersections come from the case k = 1. Therefore, the ellipse obtained
in the general case is simply a scalar multiple of a boost intersection.
Scalar multiplication preserves similarity, and so every slice intersection
is geometrically similar to a unique boost intersection. Now I will interpret
the focus-locus and focus-directrix properties, not for the slice intersections
,
but for their orthogonal projections under
to
in the plane perpendicular to the axis of the cone. The physical picture will
be simplest when we study a boost intersection as above, but it will be fairly
easy to extend to all slice intersections.
Since
the focus-locus and focus-directrix properties of ellipses are properties of
their similarity classes, it will be enough to establish and interpret
it for the projected ellipses:
considered in the special case just studied. In order to do that, we must now
recognize that the ellipses
that arise in this case (and in general, in fact) are "conic intersections."
For that, I will develop the hyperbolic geometry I need on the next page: Hyperbolic
Geometry of 2+1 Spacetime.
It is easy to see that for a general point
|
|
that is,
|
|
the construction 
gives an ellipse whose eccentricity is
|
|
since scalar multiplication
by 
transforms 
and the latter gives a boost intersection with
|
|
On
the next page, I will establish a few simple facts in hyperbolic geometry that
will eventually allow us to give a dynamic interpretation to these constructions.
