CircleBundleTransformationHyperbola

1. Transforming lines to hyperbolas

Given two points {A(r,0),B(-r,0)}, taken on the x-axis symmetrically to the origin and a direction e(cos(u),sin(u)) define a transformation as follows:
- For each point X not lying on the x-axis consider the circle member c_X(x,y) of the bundle (of all circles through the two points) generated by the circle (x²+y²)-r²=0 and the line y=0.
- Then construct Y to be the other intersection point of c_X with the line {X+te} through X and parallel to e.
The transformation Y=F(X) is well defined for every point of the plane except the x-axis. It is involutive (F² = 1) and maps non-horizontal lines of the plane to hyperbolas. This was proved in CircleBundleTransformation.html .
Here are some geometric properties of the hyperbolas thus defined.

[1] Let the line (v) be described through a vector equation X=a+tb, with (a=(a₁,0)) on the x-axis and (b) a unit vector. Then the hyperbola h=F(v) passes through {A,B} and has one asymptote passing through point C=a in the direction (e).
[2] The other asymptote of the hyperbola (h) passes through D(-a₁,0) and is inclined to the previous one by an angle equal to the angle of the line (v) to the x-axis.

Writting Y=(x,y) we saw in the aforementioned reference that the image h=F(v) of a line (v) under F may be represented in the form:
f(x,y)*g(x,y)=c, where f(x,y)=0 and g(x,y)=0 represent lines.
The claims follow by analyzing these line equations. In fact, with the notations introduced there the equation of (h) was:

The cosinus of the inclination of this line to line t=0, wich has normal (Je) results by multiplying the two unit normals, which gives b₁. This proves the second claim about the angle of the asymptotes. From the form of the equation follows that t=f(x,y)=0 and g(x,y)=0 are the two asymptotes. Obviously the first passes through (a₁,0).
That g(x,y)=0 is satisfied by (-a₁,0) is also a simple calculation.
To show that the hyperbola passes also through points {A,B} is also easy using a geometric argument. For example finding a circle bundle member such that the corresponding Y coincides with B.

Remark-1 Fixing the line a+tb and varying the direction of (e) creates various such hyperbolas, all of them passing through {A,B}. The triangle intercepted on the asymptotes by line AB has a fixed basis along AB and the opposite angle is constant. Hence the center E of the hyperbola moves on a fixed circle (c) and one of the axes of the hyperbola passes through a fixed point H on this circle. Circle (c) is tangent to line (v) at (a₁,0).

Remark-2 Changing the value of a in (a₁,0) produces hyperbolas with the same angle of asymptotics. Moving (a₁,0) so that it obtains a position inside AB changes the shape of the hyperbolas and delivers the conjugates of those for which (a₁,0) is outside AB.

Remark-3 The location of the focal points of the hyperbola can be also easily determined from the given data, using the product CB*BD and the angles of triangle CDE (see HyperbolaAsymptotics.html ).

All this works for lines (v) intersecting the x-axis. For lines (v) parallel to the x-axis the resulting image F(v) is a parabola see CircleBundleTransformationParabola.html .

2. The same angle of asymptotes

Every hyperbola can be represented in an infinite variety of ways as image F(v) for an appropriate transform F of the kind described above. For this it suffices to take a chord AB of the hyperbola and define the bundle of circles through {A,B}. From the discussion above follows that the intersection points Y of the circles of the bundle with the hyperbola define a line (v) through the inverse procedure of that used above: From Y draw a parallel to an asymptote and find the other intersection point X with the circle. For Y varying on the hyperbola, X describes a line etc..
Since there are two asymptotes, selecting one or the other defines two lines {v, v'} and two corresponding directions {e, e'} which by the procedure described above generate the hyperbola.
When points {A,B} tend to coincide then CD becomes tangent to the hyperbola, point O being its contact point and the circle-bundle becoming one of circles tangent to segment CD at its middle. The respective figure for hyperbolas is to be found below in section 5. The case of parabolas is discussed in CircleBundleTransformationParabola2.html .
Notice that all hyperbolas having the same asymptotes lines {EC,ED} belong to the same family of conics represented by equations of the form f(x,y)*g(x,y)-c=0. Here {f(x,y)=0, g(x,y)=0} describes the two lines and c is a variable parameter. All these conics are hyperbolas similar to each other and with the same eccentricity.
Finally the analogous transformation F generated by circle-bundles of non-intersecting type has similar properties. This is discussed in the file CircleBundleTransformationHyperbola2.html .

3. The parallelity of chords

Every bundle member c_X intersects line (v) in another point X', which by the definition of F maps to Y'=F(X') to a point of the same circle c_X and so that XYY'X' is a trapezium inscribed in c_X. It follows that chords YY' have a fixed direction which is the reflected direction of line (v) with respect to the direction Je of the orthogonal to vector (e).
See [5] of PowerGeneral.html for another reason why this happens. The reasoning here though applies to the analogous F constructed with circle-bundles of non-intersecting type studied in CircleBundleTransformationHyperbola2.html .
A consequence of the parallelity of chords YY' and the above remark on the inscribed trapezium is that when {X,X'} coincide on line (v) then also {Y,Y'} coincide on the hyperbola. This means that the circle-bundle members c_X which are tangent to line (v) are also tangent to the hyperbola. Besides the direction of the tangent at such a point Y is the same with the direction of chords YY'. This leads to an easy construction of the focal points of the hyperbola by determining first one circle c_X tangent to line (v) and defining an asymptotic triangle of the hyperbola (see AsymptoticTriangle.html ).

4. The tangent circles

A consequence of the parallelity of chords YY' and the above remark on the inscribed trapezium is that when {X,X'} coincide on line (v) then also {Y,Y'} coincide on the hyperbola. This means that the circle-bundle members c_X which are tangent to line (v) are also tangent to the hyperbola. Besides the direction of the tangent at such a point Y is the same with the direction of chords YY'.
Denote by {c₀,c₁} the two circle-bundle members tangent to line (v) and by {X₀,X₁} their contact points with (v). It follows that the corresponding points Y₀=F(X₀), Y₁=F(X₁) are tangent points of the hyperbola with the circles {c₀,c₁} correspondingly and that line Y₀Y₁ passes through the center E of the hyperbola and the intersection point of the line of centers OH of the bundle and the line (v). Note that by the equality of angles at E and C, circle (c) is also tangent to line (v). Line Y₀Y₁ is the conjugate direction to the direction of the parallel chords XX'.
Note that circles {c₀,c₁} do not depend on the direction (e). Thus changing this direction but leaving the same line (v) changes the hyperbolas so that they always are tangent to {c₀,c₁}.
Since the tangents at {Y₀,Y₁} are also in known directions we can easily construct the focal points of the hyperbola from a resulting asymptotic triangle of the hyperbola (see AsymptoticTriangle.html ).

5. The case of tangential bundles

The case of tangential bundle of circles tangent to line AB at its middle O is illustrated below. The resulting hyperbola has triangle ABC as tangential. The circles of the bundle are all tangent to AB at O and intersect the hyperbola in parallel chords YY', with {Y=F(X), Y'=F(X')} and {X,X'} are intersection points of the same circle with the line (v).

1. Transforming lines to hyperbolas

2. The same angle of asymptotes

3. The parallelity of chords

4. The tangent circles

5. The case of tangential bundles

See Also