La Hire theorem

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Consider a sliding plane (π) and two points D, E on this plane.
Move this plane so that D and E glide along any two fixed lines Ox and Oy.
The key property is that for any fixed point in (π), say F, the locus of F on plane (Oxy) is an ellipse : generalized La Hire theorem.
In the sequel we shall consider relative motions of (π) and (Oxy), either from the (π) point of view, that is (Oxy) moves relative to (π), or from the (Oxy) point of view, that is (π) moves relative to (Oxy).

Basic properties

Affine definition of ellipse

We recall here the ellipse is the stretch transformed of a circle.
That is the locus of point M with HM = k×HP, when P moves on circle.
Obvious analytically : x² + y² = a² is transformed by y' = (b/a)y into x²/a² + y²/b² = 1.

Paper tape method

Consider a sliding paper tape DE with D and E sliding on two perpendicular lines Ox and Oy.
A fixed point M on this tape draws an ellipse.

proof :
Let OP // EM and OP = EM = constant, then P draws a circle with radius EM.
HM/HP = HD/HO = MD/ME = constant.
Then M draws the ellipse transformed of the circle by ratio MD/ME.

If M is outside segment DE, the proof is still valid :
HM/HP = HD/HO = MD/ME = constant.

Generalized La Hire theorem

Consider plane (π) containing DE and some point F, moving with respect to plane (Oxy), with D sliding on some line Ox and E sliding on line Oy.
From the (π) point of view, O moves so that angle DOE is constant = xOy.
Hence O moves on a fixed circle in (π), centered in I. Point I is fixed in (π) and OI = constant.
Now seen from the (Oxy) point of view, point I draws a circle (as OI = constant).

 There is a point in (π) whose locus in (Oxy) is a circle. 

Consider diameter UV of circumcircle (ODE) which goes through F (that is IF line). This diameter is fixed in plane (π).
Angle UOD is constant when O moves on circle in (π), hence seen from (Oxy) point of vue, U moves on a fixed line Ou in plane (Oxy). Similarily V moves on a fixed line Ov in plane (Oxy).
As UV is a diameter, Ou is perpendicular to Ov.

 Movement of (π) is the same as if driven by U and V sliding on two perpendicular lines Ou and Ov of (Oxy) 

And as F is a fixed point on line UV, F draws an ellipse on (Oxy) : "Paper tape" UVF.
That is the generalized La Hire theorem :

 When a plane (π) moves with D and E of (π) sliding on any two lines Ox and Oy of a fixed plane, then any point F of (π) draws an ellipse. 
Note that this ellipse may degenerate into a circle or a segment.
the initial La Hire theorem is for Ox and Ox perpendicular)

Construction of locus of F

First of all we want to construct on (Oxy) the axes of this ellipse.

Set any position of DEF with D on Ox, E on Oy.
For instance with E in O and DE along Ox.
Construct the circumcircle of ODE, centered in I.
Here ODE is degenerate, hence it is the circle tangent in O to Oy,
and I is intersect of perpendicular bissector of DE and perpendicular to Oy in O.
Line IF intersects the circle in U and V, U and F on the same side from I.
Lines OU and OV are the axes of ellipse, OU being the major axis.

As OI = constant and IF = constant, the distance from O to F is maximum when O, I and F are in line.
This gives the semi major axis value = OI + IF.
The semi minor axis is |OI - IF|.

On OI draw OS = OI+IF, and OT' = |OI-IF| from a circle (I, IF).
Draw circle centered in O with radius OS. This is the major circle of ellipse.
Circle centered in O with radius OT' = OT is the minor circle.

From these axes it is easy to draw the ellipse, using the stretch property seen first.

 

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