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Elipse

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Elipse Geometry

As we have seen previously , the equation for an elipse can be written:

_ _ ~y^2 _ = _ 2 ~d ( 1 + ~e ) ~x - ( 1 - ~e^2 ) ~x^2 _ _ _ ~d >= 0 , _ 0 =< ~e =< 1

[Note: we write ~d instead of ~a in this form of the equation, as conventionally ~a represents another parameter with respect to an elipse, as we shall see below.]

~y is defined in the range _ 0 =< ~x =< 2~d ./ ( 1 - ~e ) . _ The curve is obviously symmetric about ~y = 0, i.e. about the ~x-axis. It is not difficult to show that it is also symmetric about the line ~x = ~d ./ ( 1 - ~e ) . Note that at _ ~x = ~d ./ ( 1 - ~e ) ,

~y = +- d fract{&sqrt.${1 + ~e},&sqrt.${1 - ~e}}

Elipse Equation

We will now show a simpler equation for the elipse. We'll do this by moving the curve by ~d ./ ( 1 - ~e ) to the left, i.e. so that the "centre" of the elipse co-incides with the origin.

~y^2 _ = _ 2 ~d ( 1 + ~e ) rndb{ ~x + fract{~d,1 - ~e} } _ - _ ( 1 - ~e^2 ) script{rndb{ ~x + fract{~d,1 - ~e} },,, 2,}

_ _ _ _ = _ 2 ~d _ fract{1 + ~e,1 - ~e} ( ( 1 - ~e ) ~x + ~d ) _ - _ fract{1 + ~e,1 - ~e}( ( 1 - ~e ) ~x + ~d )^2

_ _ _ _ = _ fract{1 + ~e,1 - ~e} ( 2 ~d ( 1 - ~e ) ~x + 2 ~d ^2 - (1 - ~e)^2 ~x^2 - ~d ^2 - 2 ~d (1 - ~e) ~x )

_ _ _ _ = _ fract{1 + ~e,1 - ~e} ( ~d ^2 - ( 1 - ~e )^2 ~x^2 )

so

( 1 - ~e )^2 ~x^2 + fract{1 - ~e,1 + ~e} ~y^2 _ = _ ~d ^2

which can be written

fract{~x^2,~a^2} + fract{~y^2,~b^2} _ = _ 1

where

~a^2 _ = _ fract{~d ^2,( 1 - ~e )^2 } , _ and _ ~b^2 _ = _ fract{~d ^2 ( 1 + ~e ),1 - ~e}

Dimensions

Note that ~a is just half the length of the elipse, and is known as the #~{semi major axis} of the elipse, while ~b is half the height, and known as the #~{semi minor axis}.

 

The original parameters can be expressed in terms of ~a and ~b:

fract{~b^2,~a^2} _ = _ ( 1 + ~e )( 1 - ~e ) _ = _ ( 1 - ~e^2 )

=> _ _ ~e _ = _ script{rndb{1 - fract{~b&powtwo.,~a&powtwo.}},,, 1/2 ,}

~d _ = _ ~a ( 1 - ~e )

Note also that as the elipse is symmetric about ~x = 0 , there is a second focus at _ ~x = ~e ~a , _ ~y = 0 . The directrices will be at ~d ./ ~e from the edges of the elipse, i.e. at _ ~a + ( ~a ( 1 - ~e ) ./ ~e ) _ = _ ~a ./ ~e _ from the centre.

Parametric Equation

Consider any point ~P ( ~x , ~y ) on the elipse. Draw a vertical line from the ~x-axis through ~P to the circle, radius ~a, encircling the elipse. Then draw a line from this point on the circle to the common centre of the elipse and the circle. Let the angle between this last line and the ~x-axis be &phi..

 

_ _ _ ~x _ = _ ~a cos &phi.

_ _ _ ~y^2 _ = _ ~b^2 ( 1 - ( ~x^2 / ~a^2 ) ) _ = _ ~b^2 ( 1 - cos^2 &phi. )

_ _ _ _ _ = _ ~b^2 sin^2 &phi.

~x _ = _ ~a cos &phi.
~y _ = _ ~b sin &phi.

Inter Focal Triangle

Consider an elipse with the focii at the points R and S as shown in the diagram below. Then for any point P ( ~x , ~y ) on the elipse (for illustration we show P in the top right-hand quadrant of the elipse, but the following result holds for the other quadrants by symmetry):

 

SP^2 _ = _ ( ~a~e + ~x )^2 + ~y^2 _ = _ ~a^2~e^2 + ~x^2 + ~y^2 + 2~a~e~x

but _ ~a^2~e^2 _ = _ ~a^2 ( 1 - ( ~b^2 ./ ~a^2 ) ) _ = _ ~a^2 - ~b^2

and _ ~y^2 _ = _ ~b^2 ( 1 - ( ~x^2 ./ ~a^2 ) ) _ = _ ~b^2 - ( ~b^2~x^2 ./ ~a^2 )

so _ _ SP^2 _ = _ ~a^2 + ~x^2 - ( ~b^2~x^2 ./ ~a^2 ) + 2~a~e~x _ = _ ~a^2 + ~e^2~x^2 + 2~a~e~x

Similarly _ _ RP^2 _ = _ ( ~a~e - ~x )^2 + ~y^2 _ = _ ~a^2 + ~e^2~x^2 - 2~a~e~x

 

Note that if _ ~x > ~a~e , _ as in the diagram on the right, then

RP^2 _ = _ ( ~x - ~a~e )^2 + ~y^2 , _ _ which is the same.

Now _ _ ( SP + RP )^2 _ = _ SP^2 + RP^2 + 2 SP.RP

_ _ _ = 2( ~a&powtwo. + ~e&powtwo.~x&powtwo. ) + 2 &sqrt.${ ( ~a&powtwo. + ~e&powtwo.~x&powtwo. )&powtwo. - 4 ~a&powtwo.~e&powtwo.~x&powtwo. }

_ _ _ = 2( ~a&powtwo. + ~e&powtwo.~x&powtwo. ) + 2 &sqrt.${ ( ~a&powtwo. - ~e&powtwo.~x&powtwo. )&powtwo. } _ = _ 2~a

So _ SP + RP _ = _ ~a &sqrt.$2 , _ _ for any point P on the elipse. This is a mathematical justification of the method of drawing an elipse by pinning each end of a piece of string at the focii, and keeping the string taught with the pencil to draw the elipse.