Class 38: Geometry of Conic Sections Class 38: Geometry of Conic - - PowerPoint PPT Presentation

Class 38: Geometry of Conic Sections Class 38: Geometry of Conic Sections

Orbital equation        

2 2

r L : Tangential f(r) ) r

• r

( : Radial    r L : Tangential  r 1 u  r

3 2 2 2 2 2

u L f u d u L    Solving this will give the   

2

f d  shape of the orbit k f  Special case

2

r f  Special case R S l i   cos 1 r   Solution

Conic Sections R   cos 1 R r   We will assume   0:  = 0 Circle R>0  <1 Ellipse R>0  =1 Parabola R>0  >1 Hyperbola – attractive R>0 Hyperbola – repulsive R<0

Inversion Convention

I will adopt the “inversion convention” in allowing r to be I will adopt the inversion convention in allowing r to be

• negative. If you see a negative r, you just treat it like a vector,

and do an inversion through the origin (focus).

‐r

For ellipse, parabola, and circle, the r for the whole curve have the

For R>0:

same sign, but not for hyperbola:

r 

We will use dotted line to represent the section with

Positive r Negative r

represent the section with negative r.

Ellipse, parabola, and circle Th lli (0< <1) ill l l k lik thi The ellipse (0<<1) will always look like this:

r 

Circle (=0) is a special case of ellipse when the two Circle ( 0) is a special case of ellipse when the two foci coincide at the same point (the center). Parabola (=1) is a special case when the left end of the above ellipse goes to infinite far left.

Hyperbola

Hyperbola (>1) has two branches Following our inversion Hyperbola (>1) has two branches. Following our inversion convention, one has positive r (solid line) and the other has negative r (dotted line)

r>0 r<0 r<0 r>0  r  r Case for R>0 (attractive potential) Case for R<0 (repulsive potential) ( p p )

ra and rp

1 r and r are roots of

L r k 2 r E 2 k L 1 E

2 2 2

   

• 1. ra and rp are roots of
• 2. For ellipse, parabola, and circle, ra and rp are positive.

L

• r

k 2 r E 2 r r 2 E

2

       

For ellipse, ra > rp > 0 For parabola, ra = > rp > 0 For circle, ra = rp > 0

ra rp

,

a p

• 3. For Hyperbola

r>0 r<0 r>0 r<0

(attractive potential) ra <0 , |ra| > | rp | (repulsive potential) rp<0 , |ra | > | rp |

Relationship between (R,) and (ra, rp)

      1 R cos 1 R r

• p

          1 R 180 cos 1 R r 1 cos 1

• a

   1 180 cos 1



p a p a

r r r r    

p a p a

r 2r R 

p a

r r 

Conic Sections in Cartesian Coordinates Ellipse

With the exception of parabola the  a x

Ellipse

With the exception of parabola, the

• rigin is now at the center of the conics

and the focus will have the x‐coordinate

y

f

 a x 

`

There is a line call directrix at

x

(a,0)

 a f  a x 

`

y

 x 

di t i d b t Di t focus and curve between Distance  

G l E ti ( t f b l )

x

(a 0)

directrix and curve between Distance

(Directrix of a circle is at infinite far away)

General Equation (not for parabola):

1 y x

2 2 2 2

 

(a,0)  a x 

1 b a

2 2 

Hyperbola

B is imaginary for hyperbola.  a x 

Note that a<0 for this picture

Parabola

The Cartesian Coordinate equation for parabola is y2= ‐4a’x This is not in our general form. The a’ here is not even the same “a “ as in the general equation. We can consider a parabola as part of a big ellipse as 1. However , as the ellipse becomes bigger and bigger, it center (or the origin of the Cartesian coordinates) shifts towards the left at infinity far away and that’s the problem We coordinates) shifts towards the left at infinity far away and that s the problem. We are forced to translate the coordinates by “a” to the right back to the vertex of the

• parabola. For this reason, the equation of parabola is not in pour general form:

1 b y a a 2ax Lim 1 b y a a) (x Lim

2 2 2 2 a 2 2 2 2 a    

       b Li ' h it 4 ' b y x a 2 Lim b a

2 2 2 2 a a    

   2a Lim a' h x wit

• 4a'

y

a 2  

  

Parabola y x

(a’,0)

' a x  a x 

b Lim a'

2

 a 2

b a,  

Relationship between (a,b) and (ra, rp)



p a

2 r r a  

p ar

r b 



• f

roots are r and r

p a



b ar 2 r

2 2 p

  

N t 1 b iti ti f h b l Note: 1. a can be positive or negative for hyperbola, depending on the values of ra and rp. 2 b is imaginary for hyperbola because one of r and

• 2. b is imaginary for hyperbola because one of ra and

rp(but not both) is negative.

Relationship between (a,b) and (R and )

p a

R 1 R 1 R r r a         

p a 2

R ) R )( R ( r r b 1 2 2 a         

2 p a

1 ) 1 )( 1 (      



b a

2 2 



b R a

2

  a R 

The Triangular Relationship R,

parabola for a 2 b Lim a'

2 b a,  



p a

r r r r    

2

1 R a   

R

p a p a

r r r 2r R r r   

2

1 R b 1    

b a

2 2 

    R r 1 R r

a p

p a

r r 

b R a

2

  

  1

a p a p a

r r b 2 r r a   

a

ra,rp a,b

p a

2

b ar 2 r

• f

Roots

2 2

  

Some famous triangles Ellipse y

The follow the same relationship:

2 2 2 2

a b a   

x

a a b If we allow b to be imaginary. a

y Hyperbola y

|b| |a|

x

|a|

The Triangular Relationship R,

parabola for a 2 b Lim a'

2 b a,  



E,L

p a

r r r r    

2

1 R a   

L

• r

k 2 r E 2 r k r L 2 1 E

• f

roots are r and r

2 2 2 2 p a

       

R

p a p a

r r r 2r R r r   

2

1 R b 1    

b a

2 2 

    R r 1 R r

a p

p a

r r 

b R a

2

  

  1

a p a p a

r r b 2 r r a   

a

ra,rp a,b

p a

2

b ar 2 r

• f

Roots

2 2

  

A Simplified Relationship

That’s all you need to remember, other things are simple algebra.

E,L

L k 2 E 2 r k r L 2 1 E

• f

roots are r and r

2 2 2 2 p a

  

R

L

• r

k 2 r E 2

2 2

    

    R r 1 R rp

p a

b 2 r r a  

R, ra,rp a,b

  1 ra

p ar

r b 