Anharmonic ratio

The anharmonic ratio or birapport is a powerful tool in the geometry, in particular the projective Géométrie. The name of anharmonic ratio was created by Michel Chasles but the concept is quite former for him.

Anharmonic ratio of four points

If ABCD are four points distinct from a line (d) one birapport or anharmonic ratio calls of (has, B) and (C, D) the report/ratio of algebraic measurements according to:

\ frac {\ frac {\ overline {CA}} {\ overline {CB}}} {\ frac {\ overline {DA}} {\ overline {dB}}}

divisions are supposed to be regular. The birapport of C, D compared to has, B is $\ frac {2/1} {3/2} = \ frac {4} {3}$

divisions are supposed to be regular. The birapport of C, D compared to has, B is -1/3

Properties

This report/ratio is independent of the reference mark chosen on the line (d) and of the selected unit of length.

It is easy to see that if it one permutes, at the same time A/B and C/D, one does not modify the anharmonic ratio.

This report/ratio remains invariant for many geometrical transformations: isometry, similarities, transformation closely connected. The duality by poles and polar reciprocal preserves also the anharmonic ratio of four elements of a unidimensional structure.

There remains also invariant for homography S like the central Projection…

If C is the Barycentre of (has, a) and (B, b) and if D is that of (has, a') and (B, b') then the anharmonic ratio is

\ frac {ab'} {a' B}

What explains besides why a transformation preserving the barycentres preserves also the anharmonic ratios

Four convergent line anharmonic ratio

An important result in projective geometry stipulates that a central projection preserves the anharmonic ratio. It makes it possible to say in the figure attached that the anharmonic ratios of (has, B; C, D) and (A', B'; It Of) is equal whatever the lines which carry the series of the four points. (A demonstration is realizable by using several times the theorem of Thalès).

Since this report/ratio is independent of the secant to the four lines, this report/ratio depends only on the relative position of the four lines. It is then called anharmonic ratio of the right-hand sides

$(d_A, d_B; d_C; d_D)$

See harmonic Beam

Harmonic division

When the anharmonic ratio is equal to -1 , it is said that the four points are in harmonic division . The point D then is called combined C compared to has and B. One can prove that C is also combined D compared to these same points.

Example 1: the harmonic continuation

The point of X-coordinate 1/3 is combined point of X-coordinate 1 compared to the points of X-coordinate 0 and 1/2.

the point of X-coordinate 1/4 is combined of that of X-coordinate 1/2 compared to the points of X-coordinate 0 and 1/3.

In a general way, the point of X-coordinate 1 (n+2) is combined point of X-coordinate 1/n compared to the points of X-coordinate 1 (n+1) and 0

One thus defines the continuation of numbers 1,1/2, 1/3, 1/4,… called harmonic continuation which one finds in music to define the harmonic Gamme

Example 2: harmonic mean

Combined of 0 compared to X and is the harmonic Moyenne there X and of there:

$\ frac {2} {\ frac {1} {X} + \ frac {1} {there}}$

Example 3: barycentre

If C is the barycentre of (has, a) and (B, b) then its combined compared to has and B is the barycentre of (has, - A) and (B, b)

Example 4: bisectrices

In a triangle ABC, the Bissectrice S interior and external exits of C divide line (AB) in two point D and E such as has, B, D, E form a harmonic divion

Example 5: theorem of Apollonius

The whole of the points M of the plan such as report/ratio MA/MB is constant is a circle of diameter such as has, B, C, D form a harmonic division.

Example 6: Polar of a point compared to two lines

Anharmonic ratio, lengths, surfaces and angles

In addition to its significance in terms of birapport directed lengths, the anharmonic ratio relates to also the angles and the surfaces directed. Indeed the surface of the various triangles such as OAB can be expressed in two ways

\ frac {1} {2} *OH*AB = \ frac {1} {2} *OA*AB*sin (\ widehat {AOB})

- from where, after simplifications of OH ² or OA*OB*OC*OD equality of the 3 birapports: lengths, surfaces and sine.

Anharmonic ratio on a circle

The property of the birapport of the sines has a consequence for 6 points coyclic ABCDMP. The angles

\ widehat {AMB}

and

\ widehat {APB}

being equal or additional, their sines are equal. The birapport of the right-hand sides M (ABCD) is equal to that of the right-hand sides P (ABCD). Consequently one can speak about the birapport of 4 points on a circle. It is shown, without the sines, in projective geometry that this property is true for conical unspecified (being given conical, if ABCDM are fixed and if P traverses the conical one, then the birapport lines P (ABCD) is constant).

One can deduce from it that the inversion of 4 aligned points, EFGH, of center M, preserves to them birapport on their cocyclic images ABCD.

Harmonic division, theorems of Ceva and Ménélaüs

The Theorem of Ceva and the Théorème of Ménélaüs are connected by a harmonic report/ratio. The two theorems imply two relations:

\ frac {\ overline {data base}} {\ overline {cd.}} \ frac {\ overline {THIS}} {\ overline {EA}} \ frac {\ overline {AF}} {\ overline {BFR}} = 1

and

\ frac {\ overline {Of B}} {\ overline {Of C}} \ cdot \ frac {\ overline {EC.}} {\ overline {EA}} \ cdot \ frac {\ overline {F}} {\ overline {BFR}} = 1

who, after simplification, carry out to: $\ frac {\ frac {\ overline {dB}} {\ overline {cd.}}} {\ frac {\ overline {Of B}} {\ overline {Of C}}} =-1$ , which expresses a harmonic division. While passing this property gives a construction of combined D compared to BC, by taking an arbitrary point has out of (BC) and an arbitrary point M on (AD).

See complete quadrilateral

Complexes

Dèf : Are

\ alpha, \ beta, \ gamma, \ delta

of the complexes two to two distinct. To them birraport is defined:

\ beta, \ gamma, \ delta = \ frac {\ alpha - \ gamma} {\ alpha - \ delta} * \ frac {\ beta - \ gamma} {\ beta - \ delta}

.
m. : Four points (of affix)

\ alpha, \ beta, \ gamma, \ delta

cocylcic or are aligned if \ beta, \ gamma, \ delta \ in \ mathbb {R} .

Random links: