projective harmonic conjugate

In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction:
:Given three collinear points , let be a point not lying on their join and let any line through meet at respectively. If and meet at , and meets at , then is called the harmonic conjugate of with respect to and .

The point does not depend on what point is taken initially, nor upon what line through is used to find and . This fact follows from Desargues theorem.

In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as .

Cross-ratio criterion

The four points are sometimes called a harmonic range (on the real projective line) as it is found that always divides the segment ''internally'' in the same proportion as divides ''externally''. That is:

$\overline:\overline = \overline:\overline \, .$

If these segments are now endowed with the ordinary metric interpretation of real numbers they will be ''signed'' and form a double proportion known as the cross ratio (sometimes ''double ratio'')

:$(A,B;C,D) = \frac \left/ \frac \right. ,$

for which a harmonic range is characterized by a value of −1. We therefore write:

:$(A,B;C,D) = \frac \times \frac = -1 .$

The value of a cross ratio in general is not unique, as it depends on the order of selection of segments (and there are six such selections possible). But for a harmonic range in particular there are just three values of cross ratio: since −1 is self-inverse – so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the harmonic cross-ratio.

In terms of a double ratio, given points on an affine line, the division ratio of a point is
$t(x) = \frac .$
Note that when , then is negative, and that it is positive outside of the interval.
The cross-ratio $(c,d;a,b) = \tfrac$ is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when , then and are harmonic conjugates with respect to and . So the division ratio criterion is that they be additive inverses.

Harmonic division of a line segment is a special case of Apollonius' definition of the circle.

In some school studies the configuration of a harmonic range is called ''harmonic division''.

Of midpoint

When is the midpoint of the segment from to , then
$t(x) = \frac = -1.$
By the cross-ratio criterion, the harmonic conjugate of will be when . But there is no finite solution for on the line through and . Nevertheless,
$\lim_ t(y) = 1,$
thus motivating inclusion of a point at infinity in the projective line. This point at infinity serves as the harmonic conjugate of the midpoint .

From complete quadrangle

Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:
: is the harmonic conjugate of with respect to and , which means that there is a quadrangle such that one pair of opposite sides intersect at , and a second pair at , while the third pair meet at and .

It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:
:...Staudt succeeded in freeing projective geometry from elementary geometry. In his , Staudt introduced a harmonic quadruple of elements independently of the concept of the cross ratio following a purely projective route, using a complete quadrangle or quadrilateral.

To see the complete quadrangle applied to obtaining the midpoint, consider the following passage from J. W. Young:
:If two arbitrary lines are drawn through and lines are drawn through parallel to respectively, the lines meet, by definition, in a point at infinity, while meet by definition in a point at infinity. The complete quadrilateral then has two diagonal points at and , while the remaining pair of opposite sides pass through and the point at infinity on . The point is then by construction the harmonic conjugate of the point at infinity on with respect to and . On the other hand, that is the midpoint of the segment follows from the familiar proposition that the diagonals of a parallelogram () bisect each other.

Quaternary relations

Four ordered points on a projective range are called harmonic points when there is a tetrastigm in the plane such that the first and third are codots and the other two points are on the connectors of the third codot. G. B. Halsted (1906) ''Synthetic Projective Geometry'', pages 15 & 16

If is a point not on a straight with harmonic points, the joins of with the points are harmonic straights. Similarly, if the axis of a pencil of planes is skew to a straight with harmonic points, the planes on the points are harmonic planes.

A set of four in such a relation has been called a harmonic quadruple.

Projective conics

A conic in the projective plane is a curve that has the following property:
If is a point not on , and if a variable line through meets at points and , then the variable harmonic conjugate of with respect to and traces out a line. The point is called the pole of that line of harmonic conjugates, and this line is called the polar line of with respect to the conic. See the article Pole and polar for more details.

Inversive geometry

In the case where the conic is a circle, on the extended diameters of the circle, harmonic conjugates with respect to the circle are inverses in a circle. This fact follows from one of Smogorzhevsky's theorems:
:If circles and are mutually orthogonal, then a straight line passing through the center of and intersecting , does so at points symmetrical with respect to .
That is, if the line is an extended diameter of , then the intersections with are harmonic conjugates.

Conics and Joachimthal's equation

Consider as the curve $C$ an ellipse given by the equation

:$\frac + \frac =1.$

Let $P(x_0,y_0)$ be a point outside the ellipse and $L$ a straight line from $P$ which meets the ellipse at points $A$ and $B$. Let $A$ have coordinates $(\xi,\eta)$. Next take a point $Q(x,y)$ on $L$ and inside the ellipse which is such that $A$ divides the line segment $PQ$ in the ratio $1$ to $\lambda$, i.e.
:$PA=\sqrt=1, \;\;\; AQ=\sqrt= \lambda$.
Instead of solving these equations for $\xi$ and $\eta$
it is easier to verify by substitution that the following expressions are the solutions, i.e.
:$(\xi,\eta)=\bigg(\frac, \frac\bigg).$
Since the point $A$ lies on the ellipse $C$, one has
:$\frac\bigg(\frac\bigg)^2 + \frac\bigg(\frac\bigg)^2 = 1,$
or
:$\lambda^2\bigg(\frac+\frac-1\bigg) + 2\lambda\bigg(\frac+\frac-1\bigg) + \bigg(\frac+\frac-1\bigg)=0.$
This equation - which is a quadratic in $\lambda$ - is called Joachimthal's equation.
Its two roots $\lambda_1,\lambda_2$, determine the positions of $A$ and $B$ in relation to $P$ and $Q$.
Let us associate $\lambda_1$ with $A$ and $\lambda_2$ with $B$. Then the various line segments are given by
:$QA=\frac(x-x_0, y-y_0), \;\; PA=\frac(x_0-x, y_0-y)$
and
:$QB=\frac(x-x_0, y-y_0), \;\; PB=\frac(x_0-x, y_0-y).$
It follows that
:$\frac\frac=\frac.$
When this expression is $-1$ , we have
:$\frac=-\frac.$
Thus $A$ divides $PQ$ ``internally´´ in the same proportion as $B$
divides $PQ$ ``externally´´.
The expression
:$\frac\frac$
with value $-1$ (which makes it self-inverse) is known as the harmonic cross ratio. With $\lambda_2/\lambda_1=-1$ as above, one has $\lambda_1+\lambda_2=0$ and hence the coefficient of $\lambda$ in Joachimthal's equation vanishes, i.e.
:$\frac+\frac-1=0.$
This is the equation of a straight line called the polar (line) of point (pole) $P(x_0,y_0)$. One can show that this polar of $P$ is the chord of contact of the tangents to the ellipse from $P$. If we put $P$ on the ellipse ($\lambda_1=0, \lambda_2=0$)
the equation is that of the tangent at $P$. One can also sho that the directrix of the ellipse is the polar of the focus.

Galois tetrads

In Galois geometry over a Galois field a line has points, where . In this line four points form a harmonic tetrad when two harmonically separate the others. The condition
:$(c, d; a, b) = -1, \ \text \ \ 2 (c d + a b) = (c + d) (a + b),$
characterizes harmonic tetrads. Attention to these tetrads led Jean Dieudonné to his delineation of some accidental isomorphisms of the projective linear groups for .

If , and given and , then the harmonic conjugate of is itself.

Iterated projective harmonic conjugates and the golden ratio

Let be three different points on the real projective line. Consider the infinite sequence of points , where is the harmonic conjugate of with respect to for . This sequence is convergent.F. Leitenberger (2016
Iterated harmonic divisions and the golden ratio
Forum Geometricorum 16: 429–430

For a finite limit we have

:$\lim_\frac=\Phi-2=-\Phi^ = -\frac,$

where $\Phi=\tfrac(1+\sqrt)$ is the golden ratio, i.e. $P_P\approx -\Phi^ P_P$ for large .
For an infinite limit we have

:$\lim_\frac=-1-\Phi =-\Phi^.$

For a proof consider the projective isomorphism

:$f(z)=\frac$

with

:$f \left ((-1)^n\Phi^ \right )=P_n.$

References

* Juan Carlos Alverez (2000
Projective Geometry
see Chapter 2: The Real Projective Plane, section 3: Harmonic quadruples and von Staudt's theorem.
* Robert Lachlan (1893
An Elementary Treatise on Modern Pure Geometry
link from Cornell University Historical Math Monographs.
* Bertrand Russell (1903) Principles of Mathematics, page 384.
*{{cite book , title=Pure Geometry
, first=John Wellesley, last=Russell, publisher=Clarendon Press, year=1905
, url=https://books.google.com/books?id=r3ILAAAAYAAJ

Projective geometry