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| Conic
Sections |
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Conics, a Family of Similarly Shaped Curves
- Properties of Conics
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Conics,
a Family of Similarly Shaped Curves - Properties of Conics |
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Dandelin's
Spheres - proof of conic sections focal properties
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Proof that conic section curve is
the ellipse
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Proof that conic section curve is the hyperbola
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Proof that conic section curve is the parabola
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Conics - a family of similarly shaped curves |
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| Conics, a
family of similarly shaped curves
- properties of conics |
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By intersecting
either of the two right circular conical surfaces (nappes) with the plane perpendicular to the
axis of
the cone the resulting intersection is a circle
c, as
is shown in the figure. |
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When the cutting plane is inclined to the axis of the cone at a greater angle than that made by
the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all
generators of a single cone, the resulting curve is the
ellipse
e.
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| Thus, the circle is a special case
of the ellipse in which the plane is perpendicular to the axis of the cone.
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If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is
the
parabola
p.
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When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone,
i.e., if the intersecting plane cuts both cones the
hyperbola
h
is generated.
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| Dandelin Spheres - proof of conic sections focal properties |
| Proof that conic section curve is
the ellipse
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| In the case when the plane
E
intersects all generators of the cone, as in down figure, it is possible
to inscribe two spheres which will touch the conical surface and the plane. |
| Upper sphere touches the cone surface in a circle
k1
and the plane at a point F1.
Lower sphere touches the cone surface in a circle
k2
and the plane at a
point
F2.
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Arbitrary chosen generating line g
intersects the
circle
k1
at a point
M, the circle
k2
at a point
N
and
the intersection curve e
at a point
P.
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We see that points,
M
and F1
are the tangency
points of the upper sphere and points, N
and F2
the
tangency points of the lower sphere of the tangents
drawn from the point P
exterior to the spheres.
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Since the segments of tangents from a point
exterior to sphere to the points of contact, are equal
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PM =
PF1
and PN =
PF2.
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And since planes of
circles k1and
k2,
are parallel, then are all corresponding generating segments
equal
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MN =
PM +
PN
is constant.
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Thus, the intersection curve is the locus of points
in the plane for which sum of distances from the two
fixed
points F1and
F2, is constant, i.e., the curve is
the ellipse.
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| The proof
is due to the French/Belgian mathematician Germinal Dandelin (1794 –
1847). |
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| Proof that conic section curve is the hyperbola
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| When the intersecting plane is inclined to the vertical
axis at a smaller angle than does the generator of the
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cone, the plane cuts both cones creating the
hyperbola
h which therefore consists of two disjoining
branches
as shows the
right figure.
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Inscribed spheres touch the plane on the same side
at points
F1
and F2
and the cone surface at circles
k1and k2.
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The generator
g
intersects the circles k1and k2, at
points, M and
N, and the intersection curve at the
point
P.
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By rotating the generator
g
around the vertex V
by
360°,
the point P
will move around and trace both
branches of the hyperbola.
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While rotating, the generator
will coincide with the plane two times and then will have common points
with the curve only at infinity.
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As the line
segments, PF1and
PM are the tangents
segments drawn from P
to the upper sphere, and the segments PF2
and
PN
are the tangents segments drawn to the lower sphere, then
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PM =
PF1
and PN =
PF2.
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Since the planes of
circles k1
and k2, are parallel,
then are all generating segments from
k1to
k2
of equal
length,
so
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MN =
PM -
PN
or PF1
-
PF2
is constant.
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Thus, the intersection curve is the locus of points in
the plane for which difference of distances from the
two fixed
points F1
and F2, is constant, i.e., the curve is
the hyperbola.
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| Proof that conic section curve is the parabola
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| When the cutting plane is parallel to any generator of one of the cones then we can insert only one sphere into
the cone which will touch the plane at the point F
and the cone surface at the circle
k.
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Arbitrary chosen generating line
g
intersects the circle k
at a point
M,
and the intersection curve p
at a point
P.
The point P lies on the circle
k' which is
parallel with the plane K as shows
the down figure.
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| By rotating the generator
g
around the vertex
V, the
point P
will move along the intersection curve.
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While the generator approaches
position to be parallel to the
plane
E,
the point
P will move far away from
F.
That shows the basic property of the
parabola that the line
at infinity is a tangent.
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The segments,
PF
and PM
belong to tangents drawn from P
to the sphere
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so, PM
= PF.
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Since planes of the circles,
k
and k'
are parallel to each other and
perpendicular to the section through
the cone
axis, and as the plane E
is parallel to the slanting edge
VB, then
the intersection
d, of planes
E
and
K,
is also
perpendicular to the section
through the cone axis.
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Thus, the perpendicular
PN from
P to the line
d,
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PN =
BA =
PM
or PF =
PN.
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Therefore, for any point
P
on the intersection curve the distance from the fixed point
F
is the same as it is from the fixed line
d, it proves that the intersection curve is the parabola.
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| Conics - a family of similarly shaped curves |
| A conic is the set of points
P
in a plane whose distances from a fixed point F
(the focus) and a fixed line d (the directrix), are in a constant ratio. This ratio named the
eccentricity e
determines the shape of the curve.
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| We can see that conics represent a family of similarly shaped curves if we write their equations in vertex
form.
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| Recall the method we used to transform equations of the ellipse and the hyperbola from standard to vertex form. We placed the vertex of the curve at the origin translating its graph.
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Thus, obtained are their vertex
equations;
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| y2 =
2px -
(p/a)x2 |
| -
the
ellipse
and the
circle |
| (for
the circle p =
a = r) |
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| y2 =
2px + (p/a)x2 |
| -
the
hyperbola |
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Using geometric interpretation of
these
equations we compare the area of the square
y2, formed by the ordinate of a
point
P(x,
y), with the
area of the rectangle
2p · x, whose
one side is the abscissa
x
of the point P
and the parameter 2p
other side, it follows that
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- for the ellipse the area of the square is smaller, than the area of the rectangle,
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- for the parabola is equal,
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- for the hyperbola the area of the square is greater than the area of the rectangle.
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| The names of curves were given as a result of the above relations,
so; |
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- the word “ellipse” (elleipyis)
in Greek means “deficiency," |
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- the word “parabola” (parabolh)
means “equality” and |
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- the word “hyperbola” (uperbolh)
means “excess.” |
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| In
the given vertex equations we can make following substitutions for; |
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- the ellipse |
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- the
circle |
p =
a = b = r
=> e
= 0 |
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- the
hyperbola |
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| Thus, the
equation of conics in vertex form is |
y2 =
2px -
(1 -
e2)x2.
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| The values of
e
define the curve the conic section makes, such that for |
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e = 0
- a circle, |
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0 < e < 1
- an ellipse, |
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e = 1
- a parabola, |
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e > 1
- a hyperbola, |
| as shows
the above figure. |
