In
projective geometry, the harmonic conjugate point of an
ordered triple of points on the
real projective line 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 .
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
In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instanc ...
(on the real projective line) as it is found that always divides the segment ''internally'' in the same proportion as divides ''externally''. That is:
:
If these segments are now endowed with the ordinary metric interpretation of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
they will be ''signed'' and form a double proportion known as the
cross ratio (sometimes ''double ratio'')
:
for which a harmonic range is characterized by a value of −1. We therefore write:
:
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 and on an affine line, the division ratio of a point is
:
Note that when , then is negative, and that it is positive outside of the interval.
The cross-ratio
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 inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
s.
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
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
of the segment from to , then
:
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,
:
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
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six ...
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
In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For inst ...
are called harmonic points when there is a
tetrastigm
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
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
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.
Although the definitio ...
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.
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
:
characterizes harmonic tetrads. Attention to these tetrads led Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonym ...
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
:
where is the golden ratio, i.e. for large .
For an infinite limit we have
:
For a proof consider the projective isomorphism
:
with
:
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
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teac ...
Historical Math Monographs.
* Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
(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