Conic sections

This presentation describes the mathematics of conical curves (circles, ellipse, parabolas, hyperbolas) obtained by intersecting a right circular conical surface and a plane..
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1. 1
CONIC SECTIONS
SOLO HERMELIN

2. 2
SOLO
Table of Contents
2. Circle
3. Ellipse
4. Parabola
5. Hyperbola
6. Conic sections – Analytic Expressions
7. Conic sections – General Description
CONIC SECTIONS
1. Conic Sections - Introduction
8. References

3. 3
SOLO
A right circular cone is a cone obtained by generators (straight lines) passing through
a circle, and the apex C that is situated on the normal to the circle plane and passing
trough the center of the circle. β is the angle between the cone axis and the generators.
CONIC SECTIONS
1. Conic Sections - Introduction

4. 4
SOLO
A right circular cone is a cone obtained by generators (straight lines) passing through
a circle, and the apex C that is situated on the normal to the circle plane and passing
trough the center of the circle. β is the angle between the cone axis and the generators.
CONIC SECTIONS
Cutting
Plane
generating a
"hyperbola"
Right
Circular
Cone
Cone
Apex
Conical
Section
C
Cone
Axis
Cutting
Plane
generating a
"parabola"
Cutting
Plane
generating a
"ellipse"
Cutting
Plane
generating a
"circle"
Cutting
Plane
generating
two
"lines"
a
b
÷
ø
ö
ç
è
æ
-=b
p
a
2
÷
ø
ö
ç
è
æ
-<b
p
a
2
÷
ø
ö
ç
è
æ
->b
p
a
2
( )0=a
÷
÷
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
ç
ç
è
æ
->
-=
-<
lines
line
po
2
2
1
2
int
2
b
p
a
b
p
a
b
p
a
P
F
F
*
Cutting
Plane
(Hyperbola
)
Right
Circular
Cone
Hyperbola
2
Branches
C
b
a
Ellipse
Parabola
Cutting
Plane
(Ellipse)
Cutting
Plane
(Circle)
Cutting
Plane
(Parabola)
By cutting the right circular conic by a plane we obtain different conic sections, as a
function of the inclination angle α of the plane relative to the base of the conic section
and the angle β between the generators and the base.
The discovery of the
Conical Sections is
attributed to the greek
Menachmus who lived
around 350 B.C..
1. Conic Sections - Introduction

5. 5
SOLO
The conical sections are:
CONIC SECTIONS
1. Circle if the cutting plane is normal to the cone axis (α=0) and is above or bellow
the apex.

6. 6
SOLO
The conical sections are:
CONIC SECTIONS
2. Ellipse if the cutting plane is inclined to the basis at an angle that falls short of the
angle between generators to the base (α<π/2-β) (in greek word elleipsis means
falls, short or leaves out.

7. 7
SOLO
The conical sections are:
CONIC SECTIONS
3. Hyperbola if the cutting plane is
inclined to the basis at an
angle that exceeds of the
angle between generators to
the base (α>π/2-β)(in greek word
hyperbole means excess.

8. 8
SOLO
The conical sections are:
CONIC SECTIONS
4. Parabola if the cutting plane is
parallel to a generator of the
right circular cone (α=π/2-β)
(in greek word parabole is the
origin of the words parabola and
parallel.

9. 9
SOLO
The conical sections are:
CONIC SECTIONS
5. A point- apex (α<π/2-β), one straight line (α=π/2-β), two straight lines (α>π/2-β),
if the cutting plane passes through the apex and intersects the cone basis.

10. 10
SOLO
The conical sections are:
CONIC SECTIONS
1. Circle if the cutting plane is normal to the cone axis (α=0) and is above or bellow
the apex.
2. Ellipse if the cutting plane is inclined to the basis at an angle that falls short of the
angle between generators to the base (α<π/2-β) (in greek word elleipsis means
falls, short or leaves out.
3. Hyperbola if the cutting plane is
inclined to the basis at an
angle that exceeds of the
angle between generators to
the base (α>π/2-β)(in greek word
hyperbole means excess.
4. Parabola if the cutting plane is
parallel to a generator of the
right circular cone (α=π/2-β)
(in greek word parabole is the
origin of the words parabola and
parallel.
5. A point- apex (α<π/2-β), one straight line (α=π/2-β), two straight lines (α>π/2-β),
if the cutting plane passes through the apex and intersects the cone basis.

11. 11
SOLO CONIC SECTIONS
Circle if the cutting plane is normal to the cone axis (α=0) and is above or bellow
the apex.
2. Circle

12. 12
SOLO CONIC SECTIONS
3 Ellipse (α < π/2-β)

13. 13
SOLO CONIC SECTIONS
3 Ellipse (α < π/2-β)
To find the properties of the ellipse let introduce two spheres, with centers on the cone
axis, inside the right circular cone, one the above the cutting plane and one bellow.
The sphere are tangent to the right
cone surfaces, along one circle each
(C1 for sphere 1 and C2 for sphere 2,
with centers on cone axis and
parallel to cone base), and tangent
to cutting plane at the points
F* (sphere 1) and F (sphere 2).
The center of the spheres are in the
plane perpendicular to the cutting plane.
They contain the cone axis, and are
the intersection of this axis with the
line bisecting one of the angles
generated between the cone
generators and intersection of
perpendicular and cutting planes.

14. 14
SOLO CONIC SECTIONS
Ellipse (α < π/2-β) (Continue – 1)
Let draw the cone generator CP (where C is the cone apex and P is any point on the
Ellipse).
Since PF* is tangent to sphere 1
and PF is tangent to sphere 2, and
since the tangent distances to a
sphere from the same points are
equal, we have:
**PQPF= PQPF=
Therefore
QQPQPQPFPF *** =+=+
Since Q* is on circle C1 and Q on
circle C2 and on the same generator
the distance Q*Q is independent on P.
Ellipse (Definition 1)
Ellipse is a planar curve, such that the sum of distances, from any point on the curve,
to two fixed points (foci) in the plane is constant.

15. 15
SOLO CONIC SECTIONS
Ellipse (α < π/2-β) (Continue – 2)
One other definition is obtained by the following construction:
The intersection between cutting plane and the
plane containing circle C1 is called directrix 1.
The intersection between cutting plane and the
plane containing circle C2 is called directrix 2.
The point M* on directrix 1 is on the normal
from P on directrix 1 (PM*
┴
directrix 1).
The point M on directrix 2 is on the normal
from P on directrix 2 (PM
┴
directrix 2).
The distance from the point P to the plane
containing circle C1 (that contains both Q*
and M*) is given by
The distance from the point P to the plane
containing circle C2 (that contains both Q
and M) is given by
bba cos*cos*sin*
**
PFPQPM
PFPQ=
==
bba coscossin PFPQPM
PFPQ=
==
From those equations we obtain:
Since for an ellipse α<π/2-β → sinα<sin(π/2-β) we have:
b
a
cos
sin
*
*
==
PM
PF
PM
PF
1
cos
sin
: <=
b
a
e e - eccentricity

16. 16
SOLO CONIC SECTIONS
Ellipse (α < π/2-β) (Continue – 3)
We obtained: 1
cos
sin
*
*
<=== e
PM
PF
PM
PF
b
a
P
F
Q*
Q
Cutting
PlaneRight
Circular
Cone
Sphere2 Tangent to
Cone at Q &
Cutting Plane at F
Sphere1 Tangent to
Cone at Q* &
Cutting Plane at F*
Cone
Appex
Circle C1
on the
Cone &
Sphere1
Circle C2
on the
Cone &
Sphere2
R1
R2
C
M*
M
Directrix
2
Directrix
1
a
b
F*
PQPF=
**PQPF=
constQQ
PQPQ
PFPF
=
=+
=+
*
*
*
b
b
a
cos
cos
sin
PF
PQ
PM
=
=
b
b
a
cos*
cos*
sin*
PF
PQ
PM
=
=
1:
cos
sin
*
*
<=== e
PM
PF
PM
PF
b
a
Ellipse (Definition 2)
Ellipse is a planar curve, such that the ratio of distances, from any point on the curve,
to a fixed point F* (focus 1) and to the line directrix1 and ratio of distances to a
second fixed point F and the second line directrix 2 (parallel to directrix1) are
constant and equal to e < 1.
The focci F* and F are between the two directrices, where F* is closer to directrix 1
and F to directrix 2.
The proof given here was supplied
in 1822 by the Belgian mathematician
Germinal P. Dandelin (1794-1847)

17. 17
SOLO CONIC SECTIONS
4. Parabola (α = π/2-β)
To find the properties of the parabola let introduce a sphere, with center on the cone
axis, inside the right circular cone, above the cutting plane.
The sphere is tangent to the right
cone surfaces, along one circle C
with center on cone axis and
parallel to cone base, and tangent
to cutting plane at point F.
Let draw the cone generator CP
(where C is the cone apex and P
is any point on the Parabola).
CP is tangent to the sphere at point
Q (on circle C).
Since PF is tangent to the sphere,
and all tangents from the same point
are equal PF = PQ.
Let perform the following
construction: The intersection between cutting plane and the plane containing circle C is called directrix.
The point M on directrix is on the normal from P on directrix (PM
┴
directrix).
The distance from the point P to the plane containing circle C (that contains both Q and M)
is given by bba coscossin PFPQPM
PFPQ=
==

18. 18
SOLO CONIC SECTIONS
Parabola (α = π/2-β) (Continue – 1)
P
F
Q
Cutting
Plane
Right
Circular
Cone
SphereTangent to
Cone at Q* &
Cuting Plane at F*
Cone
Apex
Plane
Containin
g
the Circle
Tangent to
Cone &
Sphere
Circle
Tangent to
Cone &
Sphere
b
bpa-=2/
M
Directrix
PQPF=
b
b
a
cos
cos
sin
PF
PQ
PM
=
=
e
PM
PF
=== :1
cos
sin
b
a
The distance from the point P to the plane containing circle C (that contains both Q and M)
is given by bba coscossin PFPQPM
PFPQ=
==
b
a
cos
sin
=
PM
PF
From those equations we obtain:
Since for a parabola α = π/2-β
→ sinα = sin(π/2-β)=cos β we have:
e - eccentricity
1
cos
sin
: ==
b
a
e e - eccentricity
Parabola (Definition)
Parabola is a planar curve, such that the distances, from any point on the curve, to a
fixed point (focus) and to the line directrix are equal.

19. 19
SOLO CONIC SECTIONS
5. Hyperbola (α > π/2-β)
To find the properties of the hyperbola let introduce two spheres, with centers on the cone
axis, inside the right circular cone, one the above the apex and one bellow.
The sphere are tangent to the right
cone surfaces, along one circle each
(C1 for sphere 1 and C2 for sphere 2,
with centers on cone axis and
parallel to cone base), and tangent
to cutting plane at the points
F* (sphere 1) and F (sphere 2).
The center of the spheres are in
the plane perpendicular to the cutting plane.
They contain the cone axis, and are
the intersection of this axis with the
line bisecting one of the angles
generated between the cone
generators and intersection of
perpendicular and cutting planes.

20. 20
SOLO CONIC SECTIONS
Hyperbola (α > π/2-β) (Continue – 1)
Let draw the cone generator CP (where C is the cone apex and P is any point on the
Hyperbola).
Since PF* is tangent to sphere 1
and PF is tangent to sphere 2, and
since the tangent distances to a
sphere from the same points are
equal, we have:
**PQPF= PQPF=
Therefore
*** QCQPQPQPFPF =-=-
Since Q* is on circle C1 and Q on
circle C2 and on the same generator
the distance Q*Q is independent on P.
P
F
F*
Q
Cutting
Plane
Right
Circular
Cone
Sphere1 Tangent to
Cone &
Cuting Plane at F*
Cone
Apex
Circle on
the Sphere2
& Cone
Sphere2 Tangent to
Cone &
Cuting Plane at F
Circle on
the
Sphere1
& Cone
Conical
Section
Conical
Section
C
Q*
Cone
Axis
Hyperbola (Definition 1)
Hyperbola is a planar curve, such that the difference of distances, from any point on the
curve, to two fixed points (foci) in the plane is constant. Hyperbola has two branches.

21. 21
SOLO CONIC SECTIONS
Hyperbola (α > π/2-β) (Continue – 2)
One other definition is obtained by the following
construction:
The intersection between cutting plane and the
plane containing circle C1 is called directrix 1.
The intersection between cutting plane and the
plane containing circle C2 is called directrix 2.
The point M* on directrix 1 is on the normal
from P on directrix 1 (PM*
┴
directrix 1).
The point M on directrix 2 is on the normal
from P on directrix 2 (PM
┴
directrix 2).
The distance from the point P to the plane
containing circle C1 (that contains both Q*
and M*) is given by
The distance from the point P to the plane
containing circle C2 (that contains both Q
and M) is given by
bba cos*cos*sin*
**
PFPQPM
PFPQ=
==
bba coscossin PFPQPM
PFPQ=
==
From those equations we obtain:
Since for an ellipse α>π/2-β → sinα>sin(π/2-β) we have:
b
a
cos
sin
*
*
==
PM
PF
PM
PF
1
cos
sin
: >=
b
a
e e - eccentricity
P
F
F*
Q
Cutting
Plane
Right
Circular
Cone
Sphere1 Tangent to
Cone &
Cuting Plane at F*
Cone
Apex
Sphere2 Tangent to
Cone &
Cuting Plane at F
Circle C1 on
the Sphere1
& Cone
Conical
Section
Conical
Section
Q*
C
Circle C2 on
the Sphere2
& Cone
PQPF=
**PQPF=
constQCQ
PQPQ
PFPF
=
=-
=-
*
*
*
b
b
a
cos
cos
sin
PF
PQ
PM
=
=
b
b
a
cos*
cos*
sin*
PF
PQ
PM
=
=
( )bpa ->2/
b
Directrix1
Directrix2M
M*
a
1:
cos
sin
*
*
>=== e
PM
PF
PM
PF
b
a

22. 22
SOLO CONIC SECTIONS
Hyperbola (α < π/2-β) (Continue – 3)
We obtained: 1
cos
sin
*
*
>=== e
PM
PF
PM
PF
b
a
Hyperbola (Definition 2)
Hyperbola is a planar curve, such that the ratio of distances, from any point on the
curve, to a fixed point F* (focus 1) and to the line directrix1 and ratio of distances to a
second fixed point F and the second line directrix 2 (parallel to directrix1) are
constant and equal to e > 1.
The focci F* and F are between the two directrices, where F* is closer to directrix 1
and F to directrix 2.
P
F
F*
Q
Cutting
Plane
Right
Circular
Cone
Sphere1 Tangent to
Cone &
Cuting Plane at F*
Cone
Apex
Sphere2 Tangent to
Cone &
Cuting Plane at F
Circle C1 on
the Sphere1
& Cone
Conical
Section
Conical
Section
Q*
C
Circle C2 on
the Sphere2
& Cone
PQPF=
**PQPF=
constQCQ
PQPQ
PFPF
=
=-
=-
*
*
*
b
b
a
cos
cos
sin
PF
PQ
PM
=
=
b
b
a
cos*
cos*
sin*
PF
PQ
PM
=
=
( )bpa ->2/
b
Directrix1
Directrix2M
M*
a
1:
cos
sin
*
*
>=== e
PM
PF
PM
PF
b
a

23. 23
SOLO CONIC SECTIONS
6. Conic Sections – Analytic Expressions
( ) ( ) aycxycx 2
2222
=++±+- ( ) ( )
2222
2 ycxaycx +--=++±
( ) ( ) ( )
2222222
44 ycxaycxaycx +--+-+=++
( )
22
ycxa
a
xc
+--=-
2222
2
22
22 ycxcxaxc
a
cx
++-=+-
222
2
22
2
cay
a
ca
x -=+
-
1
22
2
2
2
=
-
+
ca
y
a
x
cacab
b
y
a
x
>¬-=¬=+
222
2
2
2
2
1
caacb
b
y
a
x
<¬-=¬=-
222
2
2
2
2
1
01
2
2
2
2
=¬=+ c
a
y
a
x
ellipse
hyperbola
circle
Start with ellipse and hyperbola definitions:

24. 24
SOLO CONIC SECTIONS
Conic Sections – Analytic Expressions (Continue)
Ellipse and Hyperbola Polar Representations:
F
0
F
P
F
*F
x
y
r
c2
F
0F
P
F
x
y
r
p
2/p
Ellipse & Hyperbola Parabola
Parabola Polar Representations:
( )( ) ( )arrcr 2sincos2
0
222
0 =F-F+F-F+±
( )
222
0
2
44cos44 rararcrc +-=+F-F+
( )
( )
( ) a
c
e
e
ea
a
c
a
c
a
r =¬
F-F+
-
=
F-F+
÷
÷
ø
ö
ç
ç
è
æ
-
=
0
2
0
2
2
cos1
1
cos1
1
( )
0cos F-F-=rpr
( )
1
cos1
0
=¬
F-F+
= e
e
p
r
or

25. 25
SOLO CONIC SECTIONS
7. Conic Sections – General Description
Let perform a rotation of coordinates:
0
22
=+++++ FYEXDYCXYBXA
jj
jj
cossin
sincos
yxY
yxX
+-=
+=
( )
0cossinsincos
sincoscossincossin
cossin2cossin
cossin2sincos
2222
2222
2222
=++-++
-++-
-++
++
FEyxEyDxD
yxByBxB
yxCyCxC
yxAyAxA
jjjj
jjjjjj
jjjj
jjjj
Choose φ such that the coefficient of xy is zero:
( ) ( )0sincoscossin2
22
=--- jjjj BAC
( )
AC
B
-
=
02tanj

26. 26
SOLO CONIC SECTIONS
Conic Sections – General Description (Continue – 1)
0cossinsincos
cossincossin
cossin
sincos
0000
00
2
00
2
)
22
0
22
0
22
0
22
=++-++
+-
++
+
FEyxEyDxD
yBxB
yCxC
yAxA
jjjj
jjjj
jj
jj
( ) ( )
( ) ( ) 0cossinsincos
cossincossincossinsincos
0000
2
00)
2
0
22
000
2
0
2
=+++-+
+++-+
FyEDxED
yBCAxBCA
jjjj
jjjjjjjj
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
0
cossincossin2
cossin
cossinsincos2
sincos
cossincossin2
cossin
cossincossin
cossinsincos2
sincos
cossinsincos
2
00)
2
0
2
00
2
000
2
0
2
00
2
00)
2
0
2
00
00)
2
0
2
2
000
2
0
2
00
000
2
0
2
1
1
=+
÷
÷
ø
ö
ç
ç
è
æ
++
+
-
÷
÷
ø
ö
ç
ç
è
æ
-+
-
-
÷
÷
ø
ö
ç
ç
è
æ
++
+
++++
÷
÷
ø
ö
ç
ç
è
æ
-+
-
+-+
F
BCA
ED
BCA
ED
BCA
ED
yBCA
BCA
ED
xBCA
C
A
jjjj
jj
jjjj
jj
jjjj
jj
jjjj
jjjj
jj
jjjj
  
  
We obtain
or
or

27. 27
SOLO CONIC SECTIONS
Conic Sections – General Description (Continue – 2)
Let define
( )
000
2
0
2
1
cossinsincos: jjjj BCAA -+=
( )
00)
2
0
2
1
cossincossin: jjjj BCAC ++=
001
001
cossin:
sincos:
jj
jj
EDE
EDD
+=
-=
We can see that CACA +=+
11
1. If A
1
≠ 0 & C
1
≠ 0. We define
( )
( )
( )
( )
2
00)
2
0
2
00
2
000
2
0
2
00
1
cossincossin2
cossin
cossinsincos2
sincos
:
÷
÷
ø
ö
ç
ç
è
æ
++
+
-
÷
÷
ø
ö
ç
ç
è
æ
-+
-
-=
jjjj
jj
jjjj
jj
BCA
ED
BCA
ED
FF
If A
1
, C
1
, F
1
≠ 0 do not have the same algebraic sign, than the equation is an equation
of an ellipse, circle or hyperbola
1
1
1
2A
D
xx +=
1
1
1
2C
E
yy +=
The equation becomes: 0
1
2
11
2
11 =++ FyCxA
If sign A
1
= sign C
1
≠ sign F
1
& A
1
= C
1
→ circle
If sign A
1
= sign C
1
≠ sign F
1
& A
1
≠ C
1
→ ellipse
If sign A
1
≠ sign C
1
→ hyperbola

28. 28
SOLO CONIC SECTIONS
Conic Sections – General Description (Continue – 3)
2. If A
1
= 0 , C
1
≠ 0 & D
1
≠ 0. We define
1
2
1
1
1
2
:
D
C
E
F
xx
÷
÷
ø
ö
ç
ç
è
æ
-
+=
1
1
1
2
:
C
E
yy +=
Parabola0
11
2
11
=+xDyCThe equation becomes:
Parabola0
11
2
11 =+yExAThe equation becomes:
3. If A
1
≠ 0 , C
1
= 0 & E
1
≠ 0. We define
1
1
1
2
:
A
D
xx +=
1
1
1
1
2
:
E
A
D
F
yy
÷
÷
ø
ö
ç
ç
è
æ
-
+=

29. 29
SOLO CONIC SECTIONS
Conic Sections – General Description (Continue – 4)
The equation can be rewritten as 0
22
=+++++ FYEXDYCXYBXA
[ ] [ ] 0
1
2
1
2
1
2
1
2
1
2
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
1 =
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
=
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
++
++
++
Y
X
FED
ECB
DBA
YX
FEYDX
EYCXB
DYBXA
YX
The rotation of coordinates can be written as:
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
1100
0cossin
0sincos
1
y
x
Y
X
jj
jj
We can write:
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
=+++++
1
2
1
2
1
2
1
2
1
2
1
2
1
1
22
Y
X
FED
ECB
DBA
YXFYEXDYCXYBXA
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é -
=
1100
0cossin
0sincos
2
1
2
1
2
1
2
1
2
1
2
1
100
0cossin
0sincos
1 y
x
FED
ECB
DBA
yx jj
jj
jj
jj
[ ] 0
1
2
1
2
1
2
1
2
1
2
1
2
1
1
11
111
111
=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
= y
x
FED
ECB
DBA
yx

30. 30
SOLO CONIC SECTIONS
Conic Sections – General Description (Continue – 5)
Therefore we have
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é -
=
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
100
0cossin
0sincos
2
1
2
1
2
1
2
1
2
1
2
1
100
0cossin
0sincos
2
1
2
1
2
1
2
1
2
1
2
1
11
111
111
jj
jj
jj
jj
FED
ECB
DBA
FED
ECB
DBA
ú
û
ù
ê
ë
é
-
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ú
û
ù
ê
ë
é -
=
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
jj
jj
jj
jj
cossin
sincos
2
1
2
1
cossin
sincos
2
1
2
1
11
11
CB
BA
CB
BA
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
=
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ë
é
FED
ECB
DBA
FED
ECB
DBA
2
1
2
1
2
1
2
1
2
1
2
1
det
2
1
2
1
2
1
2
1
2
1
2
1
det
11
111
111
2
11
11
4
1
2
1
2
1
det
2
1
2
1
det BCA
CB
BA
CB
BA
-=
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ï
î
ï
í
ì
®>
®=
®<
-
circleorellipse
parabola
hyperbola
BCA
0
0
0
4
1
2
and
Finally we obtain:
The necessary conditions for different conic sections are:

31. 31
SOLO CONIC SECTIONS
References
1.Battin R.H., “An Introduction to the Mathematics and Methods of Astrodynamics”,
AIAA Education Series, AIAA, Washington. DV., 1987

32. 32
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
CONIC SECTIONS