Double ratio

Examples of double ratios ( are the associated partial ratios). The 3rd example shows 4 harmoniously lying points, see harmonic division .${\ displaystyle \ lambda _ {S} = (A, B; S), \ \ lambda _ {T} = (A, B; T)}$

In the simplest case, the double ratio in geometry is the ratio of two partial ratios . If, for example, the line is divided by a point as well as by a point into two partial lines and or and (see first example), the ratio is the (affine) double ratio in which the partial points divide the given line . The double ratio is of great importance as an invariant in central projections , because the clearer partial ratio is invariant under parallel projections, but not under central projections. A generalization leads to the definition of the double ratio for points of a projective line (that is, an affine line to which a far point is added). ${\ displaystyle \ left [AB \ right]}$${\ displaystyle S}$${\ displaystyle T}$${\ displaystyle \ left [AS \ right]}$${\ displaystyle \ left [SB \ right]}$${\ displaystyle \ left [AT \ right]}$${\ displaystyle \ left [TB \ right]}$${\ displaystyle {\ tfrac {| AS |} {| SB |}}: {\ tfrac {| AT |} {| TB |}}}$${\ displaystyle S, T}$${\ displaystyle \ left [AB \ right]}$

A special case is when the double ratio takes the value −1. In this case one speaks of a harmonious division of the route by the pair of points and says, lie harmoniously.${\ displaystyle \ left [AB \ right]}$${\ displaystyle S, T}$${\ displaystyle A, B, S, T}$

While the partial ratio of three points can still be estimated well from the position of the points, this is almost impossible for the double ratio. The double ratio has mainly theoretical significance in analytical and projective geometry (invariant in projective collineations ). In descriptive geometry, however, it is used (without calculation) to reconstruct flat figures.

Affine double ratio

For the parametric representation of a straight line

A line in the affine space can be determined with two selected vectors by ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle {\ vec {u}}, {\ vec {v}}}$

${\ displaystyle g \ colon {\ vec {x}} = {\ vec {u}} + x {\ vec {v}}, \ x \ in \ mathbb {R}}$

parameterize. For four points of a straight line, let the parameters with regard to the parametric representation of the straight line be . Then is called the ratio of the partial ratios${\ displaystyle A, B, S, T}$${\ displaystyle g \ colon {\ vec {x}} = {\ vec {u}} + x {\ vec {v}}}$${\ displaystyle a, b, s, t}$${\ displaystyle g}$ ${\ displaystyle (A, B; S), (A, B; T)}$

${\ displaystyle (A, B; S, T) _ {a}: = (A, B; S) :( A, B; T) = {\ frac {sa} {bs}}: {\ frac {ta } {bt}}}$

the affine double ratio of the points . ${\ displaystyle A, B, S, T}$

properties

If both partial points lie between (inner divisions) or both outside, the double ratio is positive, in the other cases (one partial point inside, the other outside) the double ratio is negative.${\ displaystyle S, T}$${\ displaystyle A, B}$

Harmonic point

If the double ratio is , one says lie harmoniously. See harmonic division . ${\ displaystyle -1}$${\ displaystyle A, B, S, T}$

Examples

Have the parameters so is . ${\ displaystyle A, B}$${\ displaystyle a = 0, b = 1}$${\ displaystyle (A, B; S, T) _ {a} = {\ tfrac {s} {1-s}}: {\ tfrac {t} {1-t}}}$

1. For is the double ratio (see picture in the introduction).${\ displaystyle s = 1/3, t = 3/4}$${\ displaystyle (A, B; S, T) = 1/6}$
2. If lying harmoniously, the following applies :, d. h, the. harmonic mean of the numbers is .${\ displaystyle A, B, S, T}$${\ displaystyle {\ tfrac {1} {2}} ({\ tfrac {1} {s}} + {\ tfrac {1} {t}}) = 1}$${\ displaystyle s, t}$${\ displaystyle 1}$

Double ratio

The “normal” double ratio is explained for four points on a projective straight line.

Projective straight line

A projective straight line over a body is the set of one-dimensional subspaces in a two-dimensional vector space. After selecting a base points of the projective line are then by homogeneous coordinate with given, the point with homogeneous coordinates the one-dimensional subspace ${\ displaystyle K}$${\ displaystyle K}$ ${\ displaystyle {\ vec {u}}, {\ vec {v}}}$${\ displaystyle (x: y)}$${\ displaystyle (x, y) \ in K ^ {2} \ setminus \ left \ {(0,0) \ right \}}$${\ displaystyle (x: y)}$

• ${\ displaystyle \ langle x {\ vec {u}} + y {\ vec {v}} \ rangle, \ (x, y) \ neq (0,0)}$
Projective line: homogeneous (above) and inhomogeneous (below) coordinates

and is therefore for everyone . The projective straight line can also be identified with, homogeneous coordinates are converted into inhomogeneous coordinates: corresponds to the point and the point . ${\ displaystyle (x: y) = (\ lambda x: \ lambda y)}$${\ displaystyle \ lambda \ in K \ setminus \ left \ {0 \ right \}}$${\ displaystyle P ^ {1} K}$${\ displaystyle K \ cup \ left \ {\ infty \ right \}}$${\ displaystyle (x: 1)}$${\ displaystyle x}$${\ displaystyle (1: 0)}$${\ displaystyle \ infty}$

The double ratio

For four points of a projective straight line with the associated homogeneous coordinates is called ${\ displaystyle A, B, S, T}$${\ displaystyle g}$${\ display style (a_ {1}: a_ {2}), (b_ {1}: b_ {2}), (s_ {1}: s_ {2}), (t_ {1}: t_ {2}) }$

• ${\ displaystyle (A, B; S, T): = {\ frac {\ begin {vmatrix} s_ {1} & a_ {1} \\ s_ {2} & a_ {2} \ end {vmatrix}} {\ begin {vmatrix} b_ {1} & s_ {1} \\ b_ {2} & s_ {2} \ end {vmatrix}}}: {\ frac {\ begin {vmatrix} t_ {1} & a_ {1} \\ t_ { 2} & a_ {2} \ end {vmatrix}} {\ begin {vmatrix} b_ {1} & t_ {1} \\ b_ {2} & t_ {2} \ end {vmatrix}}} = {\ frac {s_ { 1} a_ {2} -s_ {2} a_ {1}} {b_ {1} s_ {2} -b_ {2} s_ {1}}}: {\ frac {t_ {1} a_ {2} - t_ {2} a_ {1}} {b_ {1} t_ {2} -b_ {2} t_ {1}}}}$

the double ratio of . ${\ displaystyle A, B, S, T}$

Properties of the double ratio:

1. ${\ displaystyle (B, A; S, T) = {\ tfrac {1} {(A, B; S, T)}}}$(Swapping of )${\ displaystyle A, B}$
2. ${\ displaystyle (A, B; T, S) = {\ tfrac {1} {(A, B; S, T)}}}$(Swapping of )${\ displaystyle S, T}$
3. ${\ displaystyle (B, A; T, S) = (A, B; S, T)}$
4. ${\ displaystyle (S, T; A, B) = (A, B; S, T)}$
5. The double ratio is invariant to a base change (see rules for determinants).
6. If the four points are different from the far point , they can be described with homogeneous coordinates so that is. In this case the (affine) double ratio results (see above)${\ displaystyle \ infty}$${\ displaystyle a_ {2} = b_ {2} = s_ {2} = t_ {2} = 1}$
${\ displaystyle (A, B; S, T) = {\ tfrac {s_ {1} -a_ {1}} {b_ {1} -s_ {1}}}: {\ tfrac {t_ {1} -a_ {1}} {b_ {1} -t_ {1}}} \.}$
Invariance of the double ratio in central projection

Invariance of the double ratio

In a projective coordinate plane over a body, the projective collineations are those collineations that are generated from linear images. Since with suitable coordination, four collinear points can always be described in such a way that ${\ displaystyle A, B, C, D}$

${\ displaystyle A = (1: 0), \; B = (0: 1), \; C = (1: 1), \; D = (x: 1)}$

and a linear mapping leaves the factor invariant, the double ratio thus also remains invariant. ${\ displaystyle x}$${\ displaystyle (A, B; C, D) = x}$

In Descriptive Geometry , straight lines in space are projected onto a panel using a central projection. Such a central projection can be continued into a projective collineation of space and projective collineations leave the double ratio invariant. So:

• The double ratio remains invariant in a central projection. (see picture)

Double ratio of 4 copoint straight lines

To calculate the double ratio with angle

Because of the invariance of the double ratio in central projection, it can also be explained for four copoint straight lines lying in one plane:

• The double ratio of four copoint straight lines of a plane is the double ratio of the four points of a straight line intersecting the 4 straight lines (see picture).${\ displaystyle A, B, C, D}$

Since the amount of a (2 × 2) determinant is equal to twice the area of ​​the triangle that is spanned by the column vectors, and the area of ​​a triangle can be expressed by ( are sides of the triangle and the included angles, see triangle area ) describe the double ratio as follows: ${\ displaystyle {\ tfrac {1} {2}} from \ sin \ gamma}$${\ displaystyle a, b}$${\ displaystyle \ gamma}$

• ${\ displaystyle (A, B; C, D) = {\ frac {\ sin (CZA)} {\ sin (CZB)}}: {\ frac {\ sin (DZA)} {\ sin (DZB)}} = {\ frac {\ sin (\ alpha + \ beta)} {\ sin \ beta}}: {\ frac {\ sin (\ alpha + \ beta + \ gamma)} {\ sin (\ beta + \ gamma) }}}$ (see image).

(The side lengths are all shortened!)

Projective geometry

In a projective space , the double ratio can be calculated from the projective coordinates of the four collinear points; it is independent of the special choice of the coordinate system. Conversely, projective coordinates can be understood as double ratios. → See projective coordinate system .

The double ratio is an invariant of any projective mapping , i.e. that is, it retains its value when such mapping is used. This property can be seen as a distinctive feature of projective geometry . See also: Erlanger program . These connections were already known in antiquity and can be found e.g. B. at Pappos . They are the main reason why the term double ratio was even developed.

Double ratio and hyperbolic distance

The real projective line is the infinite boundary of the hyperbolic plane . The hyperbolic distance can be reconstructed from the double ratio as follows.

For two points and the hyperbolic plane, let the geodesic (uniquely determined) run through these two points and let their endpoints be at infinity. Let the horospheres running through or with the center and be the centers of the two to and tangential horospheres. Then the hyperbolic distance can be calculated by ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle \ gamma}$${\ displaystyle x, y}$${\ displaystyle C_ {x}, C_ {y}}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle x, y}$${\ displaystyle z, t}$${\ displaystyle C_ {x}}$${\ displaystyle C_ {y}}$

${\ displaystyle d_ {hyp} (A, B) = \ log (x, y; z, t)}$.

Conversely, the double ratio can be reconstructed from the hyperbolic distance using the formula

${\ displaystyle (x, y; z, t) = \ lim _ {(A, B, A ^ {\ prime}, B ^ {\ prime}) \ to (x, y, z, t)} \ exp {\ frac {1} {2}} \ left \ {d_ {hyp} (A, B) + d_ {hyp} (A ^ {\ prime}, B ^ {\ prime}) - d_ {hyp} (A , B ^ {\ prime}) - d_ {hyp} (B, A ^ {\ prime}) \ right \},}$

whereby the convergence takes place along a geodetic. ${\ displaystyle A \ to x, B \ to y, A ^ {\ prime} \ to z, B ^ {\ prime} \ to t}$

This formula allows a direct generalization of the double ratio for 4-tuples of points in infinity of any CAT (-1) -space , in particular a Hadamard manifold of negative section curvature .

history

The double ratio and its invariance among projectivities was used by Pappos in antiquity and rediscovered by Desargues around 1640 . It became a standard tool in the heyday of projective geometry in the 19th century. Cayley used it in 1859 in Sixth memoir on quantics to define a metric in projective geometry, see Hilbert metric . Felix Klein noticed in 1871 in Ueber the so-called non-Euclidean geometry that in this way the hyperbolic metric of the circular disk is obtained, see Beltrami-Klein model .