EllipseRatioPrinciple

1. Ellipse from the ratio principle

Define the ellipse as the locus of points P for which the ratio PF/PP' = e is constant (e<1). Here F is a fixed point and PP' is the distance of a variable point P from a fixed line L. F is called a focus of the ellipse and L is called a directrix of the ellipse.

From the definition follows that.
1) all points of the ellipse are inside the Apollonian circle of the segment FF₀, where F₀ the projection of F on the directrix. There are exactly two points {A,B} of this circle in common with the ellipse.
The circle is called the auxiliary circle of the ellipse. Points {A,B} are called vertices of the ellipse.
Obviously.
2) the ellipse is symmetric with respect to the axis AB.

2. Symmetry

Take coordinates (x,y) along the axis AB and L correspondingly. Define d = F₀F, P=(x,y). Then the definition condition implies that the points of the ellipse satisfy.
3)                                                 y² + (x-d)² = e²x².
This defines a quadratic equation for x.
                                                  x²(1-e²) - 2dx + (d²+y²) = 0.
The two roots x₁, x₂ for constant y have their middle at  d/(1-e²), which is the x-coordinate of the Apollonian circle (see ApollonianBundle.html but put there k=1/e). Thus it follows that.
4)    The ellipse is symmetric w.r to the parallel to the directrix from the center of the auxiliary circle.
Since the ellipse is also symmetric w.r to the axis AB it follows that.
5)    the ellipse is central-symmetric w.r. to point O.
This shows that.
6)     there exists a second focus F* symmetric to F w.r. to O and a second directrix L*, which is the symmetric of L w.r. to O.
The pair (F*, L*) has the same property as the (F, L) with respect to all points of the conic.

3. Relation to auxiliary

The defining relations PF = ePP' and PF* = ePP₁ imply.
7) PF + PF* = e(PP' + PP₁) = constant.
This constant is easily identified with |AB|=2ed/(1-e²) (see ApollonianBundle.html ).
Last quantity is called the major axis of the ellipse. Its radius is R = ed/(1-e²) and its center is at x₀ = d/(1-e²).

Using the definition again the ratio  (y/y')² = (e²x²-(x-d)²)/(R²-(x-x₀)²) is seen to be = 1-e². Thus.
8)    the ellipse is the image of the circle under the transformation f: (x,y)-->(x,gy), where g² = 1-e².
The form of this transformation implies that.
9)    the tangent to the circle at y' and the tangent to the ellipse at y pass through the same point S on AB.
Also if SQR is a secant of the circle then it maps under f to a secant SQ'R' of the ellipse and this shows that.
10)   the polars of the circle and the ellipse for points on the axis AB are the same.
In particular.
11)    the directrices L, L* are the polars respectively of the foci F,F* w.r. to the circle as well as the ellipse.

4. Basic tangent properties

If P is an arbitrary point on the ellipse draw from the focus F the orthogonal FP intersecting the directrix L at S.
12) Line SP is the tangent to the ellipse.

In fact, for any other point Q on SP and the projections Q', Q₁ respectively on the directrix L and SF, if Q were on the ellipse then it would be QF = eQQ', but it is also QQ₁/QQ' = PF/PP'=e => QQ₁ = QF.
This, by the orthogonality at Q₁ is impossible if Q is different from P. Thus line SP has precisely one point in common with the ellipse, hence it is its tangent. As a corollary we obtain also.
13) the tangent at P and the orthogonal at F of the focal radius FP intersect at a point S on the directrix.

A consequence of the last property in combination with the symmetry of the ellipse is the fact.
14) The angles FPQ, F*PS formed with the tangent and the focal radii at a point of the ellipse are equal.
The reason for this is the fact that quadrangles PFQP' and PF*P₁S are cyclic and these two angles are equal respectively to FP'F₀, F*P₁F₁, which by symmetry are equal.
Related to this property is also the other focal property of the ellipse by which.
15) the angles FPA, F*PB between the focal radii PF, PF* and the tangents PA, PB from P respectively are equal.

In fact, define the reflexions S, T respectively of F, F* w.r. to the tangent PA and also the reflexion Q of F* with respect to the other tangent PB from P.
Triangles PTF and PQF are equal having three equal corresponding sides.
Thus, angles  TPF  and FPQ are equal, hence angle(FPF* + 2APF) = angle(FPF* + 2F*PB).
Which shows that angle(APF) = angle(F*PB).
A consequence of the last property is a couple of properties relating to the auxiliary circle.
16)    The portion SR of the tangent at a point X between the tangents at the vertices is viewed from a focus F under a right angle.
17)    The projection P of the focus on a tangent is a point of the auxiliary circle.

The first property results by projecting F to P on the tangent at X.
Quadrangles FPST, FPRQ are then cyclic. By the previous property angle(TPF)=angle(QPR) and by the cyclic quadrangles this comes to angle(TSF)=angle(RFQ) thereby proving that angle(SFR) is a right one.
Also angle(TPQ) = angle(TPF)+angle(FPQ) = angle(TSF)+angle(FRQ) shows the other statement.
A consequence of this property is that.
18)    lines FS, FR are the bisectors of the angle(TFX).
This is seen by observing that the point of tangency X and the intersection point V of SR separate harmonically points (S,R). This in turn is seen by projecting X to point Y on the auxiliary with a parallel WX to the directrix. By circle properties (V,W) separate harmonically (T,Q), hence (V,X) separate harmonically (S,R). Thus the bundle of lines at F: F(V,S,X,R) is harmonic and two of its lines are orthogonal, hence they are bisectors of the other two (see Harmonic_Bundle.html ).
Properties 16), 17) imply two other ways to generate the ellipse.
19)   The ellipse is the envelope of the hypotenuse of the triangle SFR formed by intersecting the legs of a right-angle SFR with two fixed parallels TS, QR while turning the angle about the fixed point F lying between the parallels.
20)   The ellipse is the envelope of lines PR drawn orthogonally to segment FP at P for fixed F and P moving on a circle containing point F.

5. Standard coordinates

Making the coordinate change x'=x-x₀, y'=y, where x₀ = d/(1-e²) (see section-3), we obtain the standard form of the ellipse.
21) x'²/a² + y'²/b² = 1, with a = ed/(1-e²) and b = a*sqrt(1-e²).

1. Ellipse from the ratio principle

2. Symmetry

3. Relation to auxiliary

4. Basic tangent properties

5. Standard coordinates

See Also