1

CONIC SECTIONS

SOLO HERMELIN

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

ConeApex C

ConeAxis

Generators

BaseCircle

SOLO

A right circular cone is a cone obtained by generators (straight lines) passing througha circle, and the apex C that is situated on the normal to the circle plane and passingtrough the center of the circle. β is the angle between the cone axis and the generators.

CONIC SECTIONS1. Conic Sections - Introduction

4

SOLO

A right circular cone is a cone obtained by generators (straight lines) passing througha circle, and the apex C that is situated on the normal to the circle plane and passingtrough the center of the circle. β is the angle between the cone axis and the generators.

CONIC SECTIONS

CuttingPlane

generating a"hyperbola"

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"parabola"

CuttingPlane

generating a"ellipse"

CuttingPlane

generating a"circle"

CuttingPlane

generatingtwo

"lines"

2

2

2

0

lines

line

po

22

12

int2

P

F

F*

CuttingPlane

(Hyperbola)

RightCircular

Cone

Hyperbola2

Branches

C

Ellipse

Parabola

CuttingPlane

(Ellipse)

CuttingPlane

(Circle)

CuttingPlane

(Parabola)

By cutting the right circular conic by a plane we obtain different conic sections, as afunction of the inclination angle α of the plane relative to the base of the conic sectionand the angle β between the generators and the base.

The discovery of theConical Sections isattributed to the greekMenachmus who livedaround 350 B.C..

1. Conic Sections - Introduction

5

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"circle"

0

RightCircular

Cone

C

CuttingPlane

(Circle)

2

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

SOLO

The conical sections are:

CONIC SECTIONS

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"ellipse"

2

C

Ellipse

CuttingPlane

(Ellipse)

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

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.

CuttingPlane

generating a"hyperbola"

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

2

P

F

F*

CuttingPlane

(Hyperbola)

RightCircular

Cone

Hyperbola2

Branches

C

8

SOLO

The conical sections are:

CONIC SECTIONS

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"parabola"

2

C

Parabola

CuttingPlane

(Parabola)

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

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.

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generatingtwo

"lines"

lines

line

po

22

12

int2

C

CuttingPlane

generatingtwo

"lines"

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.

CuttingPlane

generating a"hyperbola"

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"parabola"

CuttingPlane

generating a"ellipse"

CuttingPlane

generating a"circle"

CuttingPlane

generatingtwo

"lines"

2

2

2

0

lines

line

po

22

12

int2

P

F

F*

CuttingPlane

(Hyperbola)

RightCircular

Cone

Hyperbola2

Branches

C

Ellipse

Parabola

CuttingPlane

(Ellipse)

CuttingPlane

(Circle)

CuttingPlane

(Parabola)

11

SOLO CONIC SECTIONS

RightCircular

Cone

ConeApex

ConicalSection

C

ConeAxis

CuttingPlane

generating a"circle"

0

RightCircular

Cone

C

CuttingPlane

(Circle)

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

2. Circle

12

SOLO CONIC SECTIONS

3 Ellipse (α < π/2-β)

P

FCuttingPlane Right

CircularCone

ConeAppex

C

F*

13

SOLO CONIC SECTIONS

3 Ellipse (α < π/2-β)

P

FCuttingPlane Right

CircularCone

Sphere2 Tangent toCone on Circle C2

&Cutting Plane at F

Sphere1 Tangent toCone on Circle C1

&Cutting Plane at F*

ConeAppex

Circle C1on the

Cone &Sphere1

Circle C2on the

Cone &Sphere2

R1

R2

C

F*

To find the properties of the ellipse let introduce two spheres, with centers on the coneaxis, inside the right circular cone, one the above the cutting plane and one bellow. The sphere are tangent to the rightcone 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 theline bisecting one of the angles generated between the conegenerators and intersection of perpendicular and cutting planes.

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 theEllipse).

P

F

Q*

Q

CuttingPlane Right

CircularCone

Sphere2 Tangent toCone at Q &

Cutting Plane at F

Sphere1 Tangent toCone at Q* &

Cutting Plane at F*

ConeAppex

Circle C1on the

Cone &Sphere1

Circle C2on the

Cone &Sphere2

R1

R2

C

F*

Since PF* is tangent to sphere 1and PF is tangent to sphere 2, andsince the tangent distances to a sphere from the same points areequal, we have:

** PQPF PQPF

Therefore

QQPQPQPFPF ***

Since Q* is on circle C1 and Q oncircle 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

SOLO CONIC SECTIONS

Ellipse (α < π/2-β) (Continue – 2)

P

F

Q*

Q

CuttingPlane Right

CircularCone

Sphere2 Tangent toCone at Q &

Cutting Plane at F

Sphere1 Tangent toCone at Q* &

Cutting Plane at F*

ConeAppex

Circle C1on the

Cone &Sphere1

Circle C2on the

Cone &Sphere2

R1

R2

C

M*

MDirectrix

2

Directrix1

F*

One other definition is obtained by the following construction: The intersection between cutting plane and theplane containing circle C1 is called directrix 1. The intersection between cutting plane and theplane 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 planecontaining circle C1 (that contains both Q*and M*) is given by

The distance from the point P to the planecontaining circle C2 (that contains both Qand M) is given by

cos*cos*sin***

PFPQPMPFPQ

coscossin PFPQPMPFPQ

From those equations we obtain:

Since for an ellipse α<π/2-β → sinα<sin(π/2-β) we have:

cos

sin

*

*

PM

PF

PM

PF

1cos

sin:

e e - eccentricity

16

SOLO CONIC SECTIONS

Ellipse (α < π/2-β) (Continue – 3)

We obtained: 1cos

sin

*

* e

PM

PF

PM

PF

P

F

Q*

Q

CuttingPlane Right

CircularCone

Sphere2 Tangent toCone at Q &

Cutting Plane at F

Sphere1 Tangent toCone at Q* &

Cutting Plane at F*

ConeAppex

Circle C1on the

Cone &Sphere1

Circle C2on the

Cone &Sphere2

R1

R2

C

M*

MDirectrix

2

Directrix1

F*

PQPF ** PQPF

constQQ

PQPQ

PFPF

*

*

*

cos

cos

sin

PF

PQ

PM

cos*

cos*

sin*

PF

PQ

PM

1:cos

sin

*

* e

PM

PF

PM

PF

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 1and F to directrix 2.

The proof given here was supplied in 1822 by the Belgian mathematicianGerminal P. Dandelin (1794-1847)

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 coneaxis, inside the right circular cone, above the cutting plane. The sphere is tangent to the rightcone surfaces, along one circle Cwith center on cone axis and parallel to cone base, and tangent to cutting plane at point F.

P

F

CuttingPlane

RightCircular

Cone

ConeApex

2/

P

F

Q

CuttingPlane

RightCircular

Cone

SphereTangent toCone at Q* &

Cuting Plane at F*

ConeApex

PlaneContainin

gthe CircleTangent to

Cone &Sphere

CircleTangent to

Cone &Sphere

2/

M Directrix

PQPF

cos

cos

sin

PF

PQ

PM

ePM

PF :1

cos

sin

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 pointare 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 coscossin PFPQPM

PFPQ

18

SOLO CONIC SECTIONS

Parabola (α = π/2-β) (Continue – 1)

P

F

Q

CuttingPlane

RightCircular

Cone

SphereTangent toCone at Q* &

Cuting Plane at F*

ConeApex

PlaneContainin

gthe CircleTangent to

Cone &Sphere

CircleTangent to

Cone &Sphere

2/

M Directrix

PQPF

cos

cos

sin

PF

PQ

PM

ePM

PF :1

cos

sin

The distance from the point P to the plane containing circle C (that contains both Q and M) is given by coscossin PFPQPM

PFPQ

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

1cos

sin:

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

SOLO CONIC SECTIONS

5. Hyperbola (α > π/2-β) To find the properties of the hyperbola let introduce two spheres, with centers on the coneaxis, inside the right circular cone, one the above the apex and one bellow. The sphere are tangent to the rightcone 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 theline bisecting one of the angles generated between the conegenerators and intersection of perpendicular and cutting planes.

P

F

F*

CuttingPlane

RightCircular

Cone

ConeApex

ConicalSection

ConicalSection

C

2/

P

F

F*

Q

CuttingPlane

RightCircular

Cone

Sphere1 Tangent toCone &

Cuting Plane at F*

ConeApex

Sphere2 Tangent toCone &

Cuting Plane at F

Circle C1 onthe Sphere1

& Cone

ConicalSection

Q*

CCircle C2 onthe Sphere2

& Cone

2/

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 theHyperbola).

Since PF* is tangent to sphere 1and PF is tangent to sphere 2, andsince the tangent distances to a sphere from the same points areequal, we have:

** PQPF PQPF

Therefore

*** QCQPQPQPFPF

Since Q* is on circle C1 and Q oncircle C2 and on the same generator the distance Q*Q is independent on P.

P

F

F*

Q

CuttingPlane

RightCircular

Cone

Sphere1 Tangent toCone &

Cuting Plane at F*

ConeApex

Circle onthe Sphere2

& Cone

Sphere2 Tangent toCone &

Cuting Plane at F

Circle onthe

Sphere1& Cone

ConicalSection

ConicalSection

C

Q*

ConeAxis

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

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 theplane containing circle C1 is called directrix 1. The intersection between cutting plane and theplane 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 planecontaining circle C1 (that contains both Q*and M*) is given by

The distance from the point P to the planecontaining circle C2 (that contains both Qand M) is given by

cos*cos*sin***

PFPQPMPFPQ

coscossin PFPQPMPFPQ

From those equations we obtain:

Since for an ellipse α>π/2-β → sinα>sin(π/2-β) we have:

cos

sin

*

*

PM

PF

PM

PF

1cos

sin:

e e - eccentricity

P

F

F*

Q

CuttingPlane

RightCircular

Cone

Sphere1 Tangent toCone &

Cuting Plane at F*

ConeApex

Sphere2 Tangent toCone &

Cuting Plane at F

Circle C1 onthe Sphere1

& Cone

ConicalSection

ConicalSection

Q*

CCircle C2 onthe Sphere2

& Cone

PQPF ** PQPF

constQCQ

PQPQ

PFPF

*

*

*

cos

cos

sin

PF

PQ

PM

cos*

cos*

sin*

PF

PQ

PM

2/

Directrix1

Directrix2M

M*

1:cos

sin

*

* e

PM

PF

PM

PF

22

SOLO CONIC SECTIONS

Hyperbola (α < π/2-β) (Continue – 3)

We obtained: 1cos

sin

*

* e

PM

PF

PM

PF

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 1and F to directrix 2.

P

F

F*

Q

CuttingPlane

RightCircular

Cone

Sphere1 Tangent toCone &

Cuting Plane at F*

ConeApex

Sphere2 Tangent toCone &

Cuting Plane at F

Circle C1 onthe Sphere1

& Cone

ConicalSection

ConicalSection

Q*

CCircle C2 onthe Sphere2

& Cone

PQPF ** PQPF

constQCQ

PQPQ

PFPF

*

*

*

cos

cos

sin

PF

PQ

PM

cos*

cos*

sin*

PF

PQ

PM

2/

Directrix1

Directrix2M

M*

1:cos

sin

*

* e

PM

PF

PM

PF

23

SOLO CONIC SECTIONS

6. Conic Sections – Analytic Expressions

aycxycx 22222 2222 2 ycxaycx

2222222 44 ycxaycxaycx

22 ycxaa

xc

22222

22

22 ycxcxaxca

cx

2222

222 cay

a

cax

122

2

2

2

ca

y

a

x

FF*

P M

x

yr

0

a

b

directrix2directrix

1

e

ax

e

ax

x=-c x=c

Ellipse

F*

directrix2

directrix1

F

P

x

yx=cx=-c

e

ax

e

ax

Hyperbola

a

Circle

r

Parabola

px

yy

x

directrix

M

F

cacabb

y

a

x 222

2

2

2

2

1

caacbb

y

a

x 222

2

2

2

2

1

012

2

2

2

ca

y

a

x

ellipse

hyperbola

circle

Start with ellipse and hyperbola definitions:

24

SOLO CONIC SECTIONS

Conic Sections – Analytic Expressions (Continue)

Ellipse and Hyperbola Polar Representations:

0

P

F*F

x

y

r

c2

0

P

Fx

y

r

p

2/p

Ellipse & Hyperbola Parabola

Parabola Polar Representations:

arrcr 2sincos2 0222

0

2220

2 44cos44 rararcrc

a

ce

e

ea

a

c

a

ca

r

0

2

0

2

2

cos1

1

cos1

1

0cos rpr

1cos1 0

ee

pr

or

25

SOLO CONIC SECTIONS

7. Conic Sections – General Description

Let perform a rotation of coordinates:

022 FYEXDYCXYBXA

cossin

sincos

yxY

yxX

0cossinsincos

sincoscossincossin

cossin2cossin

cossin2sincos

2222

2222

2222

FEyxEyDxD

yxByBxB

yxCyCxC

yxAyAxA

Choose φ such that the coefficient of xy is zero:

0sincoscossin2 22 BAC

AC

B

02tan

26

SOLO CONIC SECTIONS

Conic Sections – General Description (Continue – 1)

0cossinsincos

cossincossin

cossin

sincos

0000

002

002

)22

022

022

022

FEyxEyDxD

yBxB

yCxC

yAxA

0cossinsincos

cossincossincossinsincos

0000

200)

20

22000

20

2

FyEDxED

yBCAxBCA

0

cossincossin2

cossin

cossinsincos2

sincos

cossincossin2

cossincossincossin

cossinsincos2

sincoscossinsincos

2

00)

2

0

2

00

2

000

2

0

2

00

2

00)

2

0

2

0000)

2

0

2

2

000

2

0

2

00000

2

0

2

1

1

F

BCA

ED

BCA

ED

BCA

EDyBCA

BCA

EDxBCA

C

A

We obtain

or

or

27

SOLO CONIC SECTIONS

Conic Sections – General Description (Continue – 2) Let define

0002

02

1 cossinsincos: BCAA

00)2

02

1 cossincossin: BCAC

001

001

cossin:

sincos:

EDE

EDD

We can see that CACA 11

1. If A1 ≠ 0 & C1 ≠ 0. We define

2

00)2

02

00

2

0002

02

001 cossincossin2

cossin

cossinsincos2

sincos:

BCA

ED

BCA

EDFF

If A1, C1, F1 ≠ 0 do not have the same algebraic sign, than the equation is an equationof an ellipse, circle or hyperbola

1

11 2 A

Dxx

1

11 2C

Eyy

The equation becomes: 01211

211 FyCxA

If sign A1 = sign C1 ≠ sign F1 & A1 = C1 → circle

If sign A1 = sign C1 ≠ sign F1 & A1 ≠ C1 → ellipse

If sign A1 ≠ sign C1 → hyperbola

28

SOLO CONIC SECTIONS

Conic Sections – General Description (Continue – 3)

2. If A1 = 0 , C1 ≠ 0 & D1 ≠ 0. We define 1

2

1

1

1

2:

D

C

EF

xx

1

11 2:

C

Eyy

Parabola011211 xDyCThe equation becomes:

Parabola011

2

11 yExAThe equation becomes:

3. If A1 ≠ 0 , C1 = 0 & E1 ≠ 0. We define 1

11 2:

A

Dxx

1

1

1

1

2:

E

A

DF

yy

29

SOLO CONIC SECTIONS

Conic Sections – General Description (Continue – 4)

The equation can be rewritten as 022 FYEXDYCXYBXA

0

1

2

1

2

12

1

2

12

1

2

1

1

2

1

2

12

1

2

12

1

2

1

1

Y

X

FED

ECB

DBA

YX

FEYDX

EYCXB

DYBXA

YX

The rotation of coordinates can be written as:

1100

0cossin

0sincos

1

y

x

Y

X

We can write:

1

2

1

2

12

1

2

12

1

2

1

122 Y

X

FED

ECB

DBA

YXFYEXDYCXYBXA

1100

0cossin

0sincos

2

1

2

12

1

2

12

1

2

1

100

0cossin

0sincos

1 y

x

FED

ECB

DBA

yx

0

1

2

1

2

12

1

2

12

1

2

1

1

11

111

111

y

x

FED

ECB

DBA

yx

30

SOLO CONIC SECTIONS

Conic Sections – General Description (Continue – 5) Therefore we have

100

0cossin

0sincos

2

1

2

12

1

2

12

1

2

1

100

0cossin

0sincos

2

1

2

12

1

2

12

1

2

1

11

111

111

FED

ECB

DBA

FED

ECB

DBA

cossin

sincos

2

12

1

cossin

sincos

2

12

1

11

11

CB

BA

CB

BA

FED

ECB

DBA

FED

ECB

DBA

2

1

2

12

1

2

12

1

2

1

det

2

1

2

12

1

2

12

1

2

1

det

11

111

111

2

11

11

4

1

2

12

1

det

2

12

1

det BCA

CB

BA

CB

BA

circleorellipse

parabola

hyperbola

BCA

0

0

0

4

1 2

and

Finally we obtain:

The necessary conditions for different conic sections are:

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

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

CONIC SECTIONS