Extended Features:

1. To prove the geometrical construction of an ellipse, what we have to show is the sum of distances from ellipse focuses to a single point on the ellipse is a constant. After some prelimary steps, we do show that.

2. The two focuses of an ellipse have special geometric significance as indicated above. But how can we find points of special geometric significance? Some clairvoyance is needed.

Let's see if we can find point where the polar coordinates formula for an ellipse looks especially simple. Some definitions are needed:

```  e = c/a  ,  where e ≤ 1
x'=x+c  , and so x=x'-c
x'=r*cos(θ)  ,  where r is the radial coordinate from a distance c left of the geometrical
y =r*sin(θ)       center of the ellipse
and θ is the angular coordinate
measure counterclockwise from the positive x direction,
and
ρ=r/a  , where ρ is the reduced radius and r is the dimensioned radius.  ```

With these definitions, the x-y formula becomes

```  [ρ*cos(θ)-e]**2 + (a/b)**2*[ρ*sin(θ)]**2 = 1

ρ**2*[cos(θ)**2+(a/b)**2*sin(θ)**2]**2 - ρ*[2e*cos(θ)] +(e**2-1) = 0 ```

which can be best solved by the alternate quadratic formula (which uses the quadratic coefficient only once) to get

```  ρ = (1-e**2)/[-e*cos(θ)±[cos(θ)**2*(e**2+(1-(a/b)**2)*(1-e**2))
+(a/b)**2*(1-e**2)]**(1/2)]  ,```

where the cosine term in square root expression vanishes if we choose (a/b)**2=1/(1-e**2) which indeed gives a simple polar coordinates formula for an ellipse:

`  ρ = (1-e**2)/[1-e*cos(θ)] `

where we have chosen the positive case square root expression since distance ρ must be positive. If we replace c in the above derivation by -c, we get a similar formula with +e in the denominator. Compactly, the two polar coordinates formulae are

```  ρ_(∓) = (1-e**2)/[1∓e*cos(θ)]  in reduced radius;
r_(∓) = a(1-e**2)/[1∓e*cos(θ)]  in dimensioned radius,```

where the upper/lower case is for the polar coordinates origin left/right of the x-y Cartesian coordinates origin.

We define the polar coordinates origins to be ellipse focuses.

We now also have

`  b = a(1-e**2)**(1/2)  ,  b**2=a**2-c**2,  and  c**2=a**2-b**2  .  `

Note

`  e=c/a=[1-(b/a)**2]**(1/2) `

is the ellipse eccentricity, a measure of deviation of the ellipse from circularity: e=0 for a circle and e=1 for an ellipse stretched into a line.

The minimum and maximum radii from a ellipse focus are determined from calculus to occur for θ=0 and θ=π (i.e., on the long axis of the ellipse) and are, respectively,

`  r_min=a-c=a(1-e)  and  r_max=a+c=a(1+e)  .  `

From the minimum and maximum radii, one can see that e*100 % is a useful definition of the percentage difference of an ellipse from circularity.

The conventional definition of the mean radius from an ellipse focus is the mean of the minimum and maximum radii:

`  r_mean=r_min+r_max=a  , `

which is just the semi-major axis itself. This is the usual definition of mean orbital radius for elliptical orbits in astronomy (specifically in celestial mechanics).

3. To prove that the ellipse focuses have the property that the sum of distances d from them to a single point on the ellipse is a constant, we write

`  d = r_(-) + r_(+) = [(x-c)**2+y**2]**(1/2) + [(x+c)**2+y**2]**(1/2)  , `

where we suppres the prime symbol of x' for simplicity. Now

```  r_(∓)**2 = x**2 ∓ 2x*(a**2-b**2)**(1/2) + (a**2-b**2) + b**2*[1-(x/a)**2]
= x**2*[1-(b/a)**2)] ∓ 2x*(a**2-b**2)**(1/2) + a**2
= [a ∓ x*[1-(b/a)**2)]**(1/2)]**2  , ```

where a ≥ a*[1-(b/a)**2)]**(1/2) ≥ x*[1-(b/a)**2)]**(1/2) and only the positive roots have meaning for distances. Thus,

`  d = r_(-) + r_(+) = 2a `

which is a constant: QED.