ELLIPSES
AND
ELLIPTIC CURVES
M. Ram Murty
Queen’s University
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We can parametrize the points of an ellipse in the first quadrant by f : [0,π/2] →(asint,bcost). Circumference = 4∫ π/2. 0. √a2 cos2 t + b2 ...
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ELLIPSES AND ELLIPTIC CURVES M. Ram Murty Queen’s University
Planetary orbits are elliptical
What is an ellipse?
x
a
y
b
An ellipse has two foci
From:
gomath.com/geometry/ellipse.php
How to calculate arc length
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R
2
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d x d t
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d y d t
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1 0
r
d x d t
2
d y d t
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d
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Circumference of an ellipse
W e c a n p a r a m e t r i z e t h e p o i n t s o f a n e l l i p s e i n t h e fi r s t q u a d r a n t b y f : [ 0 , π / 2 ] → ( a s i n t , b c o s t ) . C i r c u m f e r e n c e = 4 R π / 2 0 p a 2 c o s 2 t
b 2 s i n 2 t d t . O b s e r v e t h a t i f a = b , w e g e t 2 π a .
The final answer
T h e c i r c u m f e r e n c e i s g i v e n b y 2 π a F ( 1 / 2 , − 1 / 2 , 1 ; λ ) w h e r e F ( a , b , c ; z ) = P ∞ n = 0 ( a ) n ( b ) n n ! ( c ) n z n i s t h e h y p e r g e o m e t r i c s e r i e s a n d ( a ) n = a ( a
1 ) ( a
2 ) · · · ( a
n − 1 ) .
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w
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F
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4 x ( 1
x ) 2
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2 a
F
a
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b
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x
2
T h e a n s w e r c a n n o w b e w r i t t e n a s π ( a
b ) F ( − 1 / 2 , − 1 / 2 , 1 ; x 2 ) .
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e
p
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t
a
b
a
n
d
x
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Approximations
In 1609, Kepler used the approximation (a+b). The above formula shows theperimeter is always greater thanthis amount. - In 1773, Euler gave theapproximation 2 √ (a²+b²)/2. - In 1914, Ramanujan gave the approximation (3(a+b) - √ (a+3b)(3a+b)).
What does this mean?
I n 1 8 8 2 , F e r d i n a n d v o n L i n d e m a n n p r o v e d t h a t π i s t r a n s c e n d e n t a l . T h i s m e a n s t h a t π d o e s n o t s a t i s f y a n e q u a t i o n o f t h e t y p e x n
a n − 1 x n − 1
· · ·
a 1 x
a 0 = 0 w i t h a i r a t i o n a l n u m b e r s . 1 8 5 2
1 9 3 9
This means that you can’t square
the circle!
There is an ancient problem ofconstructing a square with straightedgeand compass whose area equals . - Theorem: if you can construct a linesegment of length α then α is an algebraic number. - Since is not algebraic, neither is √ .
Not both can be algebraic!
Here’s a proof. - If both are algebraic, then (x − e ) ( x − π ) = x 2 − ( e
π ) x
π e i s a q u a d r a t i c p o l y n o m i a l w i t h a l g e b r a i c c o e ffi c i e n t s . T h i s i m p l i e s b o t h e a n d π a r e a l g e b r a i c , a c o n t r a d i c t i o n t o t h e t h e o r e m s o f H e r m i t e a n d L i n d e m a n n . C o n j e c t u r e : π a n d e a r e a l g e b r a i c a l l y i n d e p e n d e n t .
But what about the case of the
ellipse?
L e t s l o o k a t t h e i n t e g r a l a g a i n . Y e s . T h i s i s a t h e o r e m o f T h e o d o r S c h n e i d e r ( 1 9 1 1
1 9 8 8 ) I s t h e i n t e g r a l R π / 2 0 p a 2 c o s 2 t
b 2 s i n 2 t d t t r a n s c e n d e n t a l i f a a n d b a r e r a t i o n a l ?