Graphics Reference
In-Depth Information
Figure 3.20.
Pencil and string construction for an
ellipse.
of fixed length to the foci, stretching the string tightly with a pencil, and drawing all
the points that can be reached by the pencil in that way. See Figure 3.20. The next
theorem expresses this geometric characterization and related ones for the hyperbola
and parabola mathematically.
3.6.2
Theorem
(1) An ellipse can be defined as a set of points such that the sum of their distances
to two fixed points is constant. The two points are the foci of the ellipse.
(2) A hyperbola can be defined as a set of points such that the difference of their
distances to two fixed points is constant. The two points are the foci of the
hyperbola.
(3) A parabola can be defined as a set of points such that the sum of their dis-
tances to a fixed point and a fixed line is constant.
Proof.
See [Eise39] or [Full73].
Definition.
Two central conic sections in the plane are said to be
confocal
if they
have the same foci.
One can show that the family of confocal ellipses for two fixed foci covers the
plane with each point in the plane belonging to a unique ellipse in the family. A similar
fact holds for confocal hyperbolas. See [HilC99]. Furthermore, if an ellipse and a
hyperbola have the same foci, then the two curves intersect orthogonally.
Definition.
The points of a conic section where it intersects the line through the
focus that is orthogonal to the directrix are called the
vertices
of that conic section.
It is easy to show that parabolas have one vertex and ellipses and hyperbolas have
two vertices. Now let
C
be a conic section and let
L
1
be the line through its focus
p
that is orthogonal to the directrix. Define a point
O
as follows: If
C
is a parabola, then
O
is its vertex, otherwise, we have two vertices and
O
is the midpoint of the segment
that has them as end points. Let
L
2
be the line through
O
orthogonal to
L
1
. The orthog-
onal lines
L
1
and
L
2
determine a natural coordinate system for our conic section. Let
u
1
=
Op
/|
Op
|, and let
u
2
be a unit direction vector for
L
2
.
Definition.
The coordinate system for the plane containing the conic section
C
determined by the frame (
u
1
,
u
2
,
O
) is called the
natural coordinate system
for the conic
