Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 2, 171-184.

Metric Properties of the Stephanos Bijection

Dedicated to Professor G. Geise on his 65^{th} birthday

Herbert Urban

Mathematisches Institut der TU München, Arcisstraße 21, D-80290 München

Abstract. The three-dimensional projective image space P^{3} of the Stephanos bijection becomes a hyperbolic space P_{|2}^{3} with index 2, if the ruled quadric Q_{42}^{2} which corresponds to the singular collineations of the projective line P^{1} is regarded as an absolute quadric. We study properties of the Stephanos bijection which are connected with the Cayley/Klein metric defined in P_{|2}^{3}.

1. Introduction

Stephanos’ bijection of the collineations of the real projective line P^{1} onto the points of the real three-dimensional projective space P^{3} has been often investigated (v. Horniaček [5], Medek [6],[7],[8], Geise [2] and the references in [2]), but all these articles are restricted to projective properties of the Stephanos bijection. If the ruled quadric Q_{42}^{2} which corresponds to the singular collineations is regarded as an absolute quadric, the P^{3} becomes a hyperbolic space P_{|2}^{3} with index 2 (cf. Giering [3], pp. 234-235). This paper examines for the first time properties of the Stephanos bijection which are connected with the Cayley/Klein metric defined in P_{|2}^{3}. The locus of the Stephanos image points P_{A} of all elliptic or all directly hyperbolic projectivities Π_{A} with a given characteristic cross ratio c_{A} is a distance sphere with the Stephanos image point P_{E} of the identity as centre. For the Stephanos image points P_{A} and P_{B} of two hyperbolic projectivities Π_{A} and Π_{B} with altogether four pairwise distinct fixed points Q_{A}, R_{A}, Q_{B} and R_{B}, the angle of the lines P_{E} + P_{A} and P_{E} + P_{B} can be given as a function of the cross ratio δ_{F} = DV (Q_{A}, R_{A}, Q_{B}, R_{B}), if Q_{A} and Q_{B} are not separated by R_{A} and R_{B} on the projective line P^{1}. If Π_{A} and Π_{B} are directly hyperbolic projectivities, the first cosine law of hyperbolic trigonometry yields a formula which expresses the distance d(P_{A}, P_{B}) by δ_{F} and the characteristic cross ratios c_{A} and c_{B} of Π_{A} and Π_{B} (Proposition 7). For three projectivities Π_{A}, Π_{B} and Π_{C} we represent the angle of the planes ε_{AB} = P_{E} + P_{A} + P_{B} and ε_{AC} = P_{E} + P_{A} + P_{C} (if it is defined) depending on the relative position of the six fixed points of the three projectivities.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

The Stephanos bijection is a kinematic mapping of the collineations of the projective line P^{1}. Rath [9] studies a kinematic mapping of the collineations of the n-dimensional projective spaces which includes the Stephanos bijection as the special case n = 1. Metric properties which are similar to the results obtained in this paper can be given especially for those kinematic mappings that have -as the Stephanos bijection- a Cayley/Klein space as image space (see for example Grünwald [4], Blaschke [1], pp. 177-258, Rehbock [10], Strubecker [12] and Röschel [11]).

2. Preliminaries

Let Π_{A} be a collineation of the real projective line P^{1} with the transformation matrix

A = ( \begin{matrix}a_{0} & a_{1} \\ a_{2} & a_{3} \end{matrix} ) .  (1)

If Π_{A} is a projectivity (det A ≠ 0) and different from the identity Π_{E} = id, then Π_{A} has two conjugate complex or real (possibly coincident) fixed points Q_{A} and R_{A} with the appertaining eigenvalues λ_{A} and μ_{A}:

\begin{matrix}λ_{A} = \frac{1}{2}(a_{0} + a_{3}) − \frac{1}{2}\sqrt{D_{A}} , μ_{A} = \frac{1}{2}(a_{0} + a_{3}) + \frac{1}{2}\sqrt{D_{A}} ; \\ D_{A} := (a_{0} − a_{3})^{2} + 4a_{1}a_{2}. \end{matrix}  (2)

For the homogeneous coordinates q_{A} and r_{A} of Q_{A} and R_{A} we have:

\begin{matrix}q_{A} = ( \begin{matrix}a_{1} \\ λ_{A} − a_{0} \end{matrix} ) , r_{A} = ( \begin{matrix}a_{1} \\ μ_{A} − a_{0} \end{matrix} ) , if a_{1} ≠ 0 , \\ q_{A} = ( \begin{matrix}λ_{A} − a_{3} \\ a_{2} \end{matrix} ) , r_{A} = ( \begin{matrix}μ_{A} − a_{3} \\ a_{2} \end{matrix} ) , if a_{2} ≠ 0 , \\ q_{A} = ( \begin{matrix}1 \\ 0 \end{matrix} ) , r_{A} = ( \begin{matrix}0 \\ 1 \end{matrix} ) or q_{A} = ( \begin{matrix}0 \\ 1 \end{matrix} ) , r_{A} = ( \begin{matrix}1 \\ 0 \end{matrix} ) , if a_{1} = a_{2} = 0 . \end{matrix} \} (3)

The projectivity Π_{A} ≠ id is called \{ \begin{matrix}elliptic \\ hyperbolic \\ parabolic \end{matrix} \} , if the fixed points Q_{A} and R_{A} are

\{ \begin{matrix}conjugate complex \\ real and distinct \\ coincident \end{matrix} \}. The characteristic cross ratio c_{A} of Π_{A} is defined as

c_{A} := \frac{λ_{A}}{μ_{A}} = DV (Q_{A}, R_{A}, X, Π_{A}X) ∀X ∈ P^{1} .  (4)

A parabolic projectivity has the characteristic cross ratio c_{A} = 1. For an elliptic projectivity, c_{A} has the form

c_{A} = e^{i·φ_{A}} ; 0 < φ_{A} < 2π.  (5)

A hyperbolic projectivity is called \{ \begin{matrix}directly \\ indirectly \end{matrix} \} hyperbolic, if \{ \begin{matrix}c_{A} > 0 \\ c_{A} < 0 \end{matrix} \}. The involutions

are characterized by the characteristic cross ratio c_{A} = −1:

Π_{A}^{2} = id ⟺ c_{A} = −1 .  (6)

Two non-parabolic projectivities Π_{A} and Π_{B} with identical fixed points Q_{A} = Q_{B} and R_{A} = R_{B} have the same characteristic cross ratio, if and only if they are inverse to each other:

Q_{A} = Q_{B} ≠ R_{A} = R_{B} : c_{A} = c_{B} ⟺ Π_{A} = Π_{B}^{−1} .  (7)

3. The Stephanos Bijection

The Stephanos bijection maps the collineations of the real projective line P^{1} onto the points of the real three-dimensional projective space P^{3}. A collineation Π_{A} with the transformation matrix A (A as in (1)) is mapped onto its Stephanos image point

P_{A} = [(a_{0}, a_{1}, a_{2}, a_{3})^{⊤}] ∈ P^{3} .  (8)

The singular collineations are mapped onto the points of a ruled quadric Q_{42}^{2} ⊂ P^{3}:

Q_{42}^{2} = {X ∈ P^{3} : x_{0}x_{3} − x_{1}x_{2} = 0}.  (9)

If the quadric Q_{42}^{2} is regarded as an absolute quadric, the projective space P^{3} becomes a hyperbolic space P_{|2}^{3} with index 2 (see [3], pp. 234-235). The points X ∈ P^{3}\Q_{42}^{2} are called the finite points, the points X ∈ Q_{42}^{2} are called the infinite points of P_{|2}^{3}. The Stephanos image point P_{E} of the identity Π_{E} = id is

P_{E} = [(1, 0, 0, 1)^{⊤}] .  (10)

The polar plane Γ_{E} of P_{E} with respect to Q_{42}^{2} has the equation

Γ_{E} = {X ∈ P^{3} : x_{0} + x_{3} = 0}.  (11)

Proposition 1. The Stephanos bijection maps the projectivities of P^{1} with a given (real or complex) fixed point Q onto the finite points of a (real or complex) tangent plane of the absolute quadric Q_{42}^{2} which contains the Stephanos image point P_{E} of the identity.

Proof: If λ is the eigenvalue of a fixed point Q = [(q_{0}, q_{1})^{⊤}] of a collineation Π_{A} (A as in (1), then we get from

a_{0}q_{0} + a_{1}q_{1} = λq_{0} , a_{2}q_{0} + a_{3}q_{1} = λq_{1}  (12)

by elimination of λ:

q_{0}q_{1}(a_{0} − a_{3}) + q_{1}^{2}a_{1} − q_{0}^{2}a_{2} = 0 .  (13)

If q_{0} and q_{1} are given (homogeneous) coordinates of the fixed point Q, (13) is the equation of a plane containing P_{E} (see (10)). The pole of this plane with respect to Q_{42}^{2} has the homogeneous coordinates (−q_{0}q_{1}, q_{0}^{2}, −q_{1}^{2}, q_{0}q_{1})^{⊤} and is therefore a point of the absolute quadric Q_{42}^{2}. Hence the plane is a tangent plane of Q_{42}^{2}.

Corollary 2. The Stephanos bijection maps the projectivities of P^{1} with two given conjugate complex or real (possibly coincident) fixed points Q and R onto the finite points of a (real) line which contains the Stephanos image point P_{E} of the identity.

Proof: If the fixed points Q and R are distinct, the assertion follows from Proposition 1. If Q and R are coincident, we get from (2) for the appertaining (double) eigenvalue λ:

λ = \frac{1}{2}(a_{0} + a_{3}) .  (14)

From (12) and (14) follows:

−q_{0}a_{0} + 2q_{1}a_{1} + q_{0}a_{3} = 0 , −q_{1}a_{0} + 2q_{0}a_{2} + q_{1}a_{3} = 0 .  (15)

If q_{0} and q_{1} are given (homogeneous) fixed point coordinates, (15) describes a line which contains the point P_{E} (see (10)).

Remark R1. Proposition 1 yields a result which was already given by Geise in [2], pp. 374-375: If Q_{31}^{2} ⊂ P_{|2}^{3} denotes the tangent cone from P_{E} to Q_{42}^{2}, then the Stephanos image point

P_{A} of a \{ \begin{matrix}elliptic \\ hyperbolic \\ parabolic \end{matrix} \} projectivity Π_{A} ≠ id is a point of \{ \begin{matrix}the interior domain of Q_{31}^{2} \\ the exterior domain of Q_{31}^{2} \\ Q_{31}^{2} \end{matrix} \}.

Proposition 3. The Stephanos bijection maps the projectivities of P^{1} with a given characteristic cross ratio c_{A} onto the finite points of a quadric Q^{2}(c_{A}) ⊂ P_{|2}^{3} which belongs to the pencil of quadrics which is spanned by Q_{42}^{2} and the doubly counted polar plane Γ_{E} of the Stephanos image point P_{E} of the identity with respect to Q_{42}^{2}.

Proof: For the eigenvalues λ_{A} and μ_{A} and the characteristic cross ratio c_{A} of a projectivity Π_{A} we get with (4):

c_{A}(λ_{A} + μ_{A})^{2} = (1 + c_{A})^{2} · λ_{A}μ_{A} .  (16)

With λ_{A} + μ_{A} = trA = a_{0} + a_{3} and λ_{A}μ_{A} = detA = a_{0}a_{3} − a_{1}a_{2} (see (1)) follows

c_{A}(a_{0} + a_{3})^{2} = (1 + c_{A})^{2}(a_{0}a_{3} − a_{1}a_{2}) .  (17)

The statement of Proposition 3 follows with (9) and (11).

Remark R2. The projectivities with the characteristic cross ratio c_{A} = −1 (the involutions) correspond to the finite points of the polar plane Γ_{E}. The projectivities with c_{A} = 1 (the parabolic projectivities) are mapped onto the finite points of the tangent cone Q_{31}^{2} from P_{E} to Q_{42}^{2} according to Remark 1. All other quadrics of the pencil of quadrics considered in Proposition 3 are non-degenerate ruled quadrics.

Remark R3. The Stephanos image points P_{A} and P_{A^{−1}} of the inverse projectivities Π_{A} and Π_{A}^{−1} with two given distinct fixed points Q_{A} and R_{A} and the given characteristic cross ratio c_{A} ∉ {−1, 1} (see equation (7)) are the intersection points of the line of P_{|2}^{3} through P_{E} corresponding to Q_{A} and R_{A} (see Corollary 2) and the ruled quadric corresponding to c_{A} (see Proposition 3). Therefore the points P_{A} and P_{A^{−1}} are mapped onto each other by the harmonic homology ^{1} with the centre P_{E} and the axis Γ_{E}. If the intersection point of the line P_{E} + P_{A} = P_{E} + P_{A^{−1}} and the polar plane Γ_{E} is denoted by S_{A}, then the points P_{A}, P_{A^{−1}}, P_{E} and S_{A} are in harmonic position (see [3], p. 72, Satz 1):

c_{A} ∉ {1, −1} : DV (P_{A}, P_{A^{−1}}, P_{E}, S_{A}) = −1 ; S_{A} := (P_{E} + P_{A}) ∩ Γ_{E} .  (18)

4. Metric Properties of the Stephanos Bijection

Proposition 4. Let Π_{A} ≠ id be a non-parabolic projectivity of P^{1}, P_{A} the Stephanos image point of Π_{A} and S_{1} and S_{2} the (real or complex) intersection points of the absolute quadric Q_{42}^{2} ⊂ P_{|2}^{3} and the connecting line of P_{A} and the Stephanos image point P_{E} of the identity. If the points S_{1} and S_{2} are numbered suitably, then the cross ratio of the four points P_{E}, P_{A}, S_{1} and S_{2} is equal to the characteristic cross ratio c_{A} of Π_{A}:

DV (P_{E}, P_{A}, S_{1}, S_{2}) = c_{A}; {S_{1}, S_{2}} = (P_{E} + P_{A}) ∩ Q_{42}^{2} .  (19)

Proof: By using the set-up

S_{i} = [α_{i}p_{A} + β_{i}p_{E}] ; i = 1, 2 ,  (20)

we get the coefficients α_{i} and β_{i} on account of (8), (9) and (10) as homogeneous solutions of the quadratic equation

(α_{i}a_{0} + β_{i})(α_{i}a_{3} + β_{i}) − α_{i}a_{1} · α_{i}a_{2} = 0 ; i = 1, 2 .  (21)

If we denote the eigenvalues appertaining to Π_{A} with λ_{A} and μ_{A} (see (2)), then from (21) follows:

(\frac{β_{i}}{α_{i}} )^{2} + (λ_{A} + μ_{A}) · \frac{β_{i}}{α_{i}} + λ_{A}μ_{A} = 0 ; i = 1, 2 .  (22)

Equation (22) has the two solutions

\frac{β_{1}}{α_{1}} = −μ_{A} , \frac{β_{2}}{α_{2}} = −λ_{A} .  (23)

1 See [3], pp. 40-41.

For the cross ratio considered in (19) we have because of (20):

DV (P_{E}, P_{A}, S_{1}, S_{2}) = \frac{β_{2}}{α_{2}} : \frac{β_{1}}{α_{1}} .  (24)

The assertion (19) follows from (23), (24) and (4).

Remark R4. Let Π_{A} and Π_{B} (Π_{A} ≠ Π_{B}) be two non-parabolic projectivities, being different from the identity, with common fixed points Q_{A} = Q_{B} and R_{A} = R_{B} and the characteristic cross ratios c_{A} and c_{B}. Let P_{A} and P_{B} be the Stephanos image points of Π_{A} and Π_{B} and S_{1} and S_{2} the intersection points of the absolute quadric Q_{42}^{2} of P_{|2}^{3} and the line P_{A} + P_{B} (which contains P_{E}). Then we have

DV (P_{A}, P_{B}, S_{1}, S_{2}) = \frac{DV (P_{E}, P_{B}, S_{1}, S_{2})}{DV (P_{E}, P_{A}, S_{1}, S_{2})} = \frac{c_{B}}{c_{A}} .  (25)

Remark R5. If Π_{A} is an elliptic projectivity, then the connecting line P_{E} + P_{A} is a passant of the hyperbolic space P_{|2}^{3} according to Remark R1, and on account of (19) the (directed) distance ^{2} d(P_{E}, P_{A}) of P_{E} and P_{A} can be written as (φ_{A} as in (5)):

d(P_{E}, P_{A}) = \frac{1}{2i} ln c_{A} = \frac{1}{2}φ_{A} .  (26)

If Π_{A} is a hyperbolic projectivity, then P_{E} + P_{A} is a secant of P_{|2}^{3} according to Remark R1. If furthermore Π_{A} is directly hyperbolic (c_{A} > 0), then P_{E} and P_{A} are not separated on P_{E} + P_{A} by the intersection points S_{1} and S_{2} of P_{E} + P_{A} and Q_{42}^{2} (see [2], p. 374, 3.). In this case we have:

d(P_{E}, P_{A}) = \frac{1}{2} ln c_{A} .  (27)

According to (26) and (27) the locus of the Stephanos image points of all elliptic or all directly hyperbolic projectivities with a given characteristic cross ratio is a distance sphere of the hyperbolic space P_{|2}^{3} with the centre P_{E}.

Proposition 5. Let Π_{A} and Π_{B} be two projectivities of P^{1}, being different from the identity, with altogether four pairwise distinct (real or complex) fixed points Q_{A}, R_{A}, Q_{B} and R_{B}. Furthermore let ε_{AB} be the plane of the hyperbolic space P_{|2}^{3} which is spanned by the Stephanos image points P_{A}, P_{B} and P_{E} of Π_{A}, Π_{B} and the identity and let g_{1} and g_{2} be the two (real or complex) tangents from P_{E} to Q_{42}^{2} which lie in ε_{AB}. Then the cross ratios

δ_{F} := DV (Q_{A}, R_{A}, Q_{B}, R_{B}) , δ_{G} := DV (P_{E} + P_{A}, P_{E} + P_{B}, g_{1}, g_{2})  (28)

are connected by

(1 − δ_{F})^{2}(1 − δ_{G})^{2} = 16 · δ_{F} δ_{G} .  (29)

2 See [3], pp. 177-178, Satz 3.

Proof: Case 1: Let all four fixed points be complex. Without loss of generality, Q_{A} and R_{A} have the homogeneous coordinates

q_{A} = ( \begin{matrix}1 \\ i \end{matrix} ) , r_{A} = ( \begin{matrix}1 \\ −i \end{matrix} ) .  (30)

Then we get:

δ_{F} = \frac{q_{B1} − iq_{B0}}{q_{B1} + iq_{B0}} : \frac{r_{B1} − ir_{B0}}{r_{B1} + ir_{B0}} .  (31)

If the entry b_{1} of the transformation matrix

B = ( \begin{matrix}b_{0} & b_{1} \\ b_{2} & b_{3} \end{matrix} )  (32)

of Π_{B} were zero, the point [(0, 1)^{⊤}] would be a fixed point in contradiction to the assumption. Therefore b_{1} ≠ 0 holds, and we get from (31) by application of (3) (λ_{B} and μ_{B} denote the eigenvalues of Π_{B}):

δ_{F} = \frac{(λ_{B} − b_{0}) − ib_{1}}{(λ_{B} − b_{0}) + ib_{1}} : \frac{(μ_{B} − b_{0}) − ib_{1}}{(μ_{B} − b_{0}) + ib_{1}} =^{(2)} \frac{(b_{1} − b_{2}) + i(λ_{B} − μ_{B})}{(b_{1} − b_{2}) − i(λ_{B} − μ_{B})} .  (33)

Let P̃_{A}, P̃_{B}, G_{1} and G_{2} be the intersection points of the four lines P_{E} + P_{A}, P_{E} + P_{B}, g_{1} and g_{2} with the polar plane Γ_{E} of P_{E} with respect to Q_{42}^{2}:

\begin{matrix}P̃_{A} := (P_{E} + P_{A}) ∩ Γ_{E} , & P̃_{B} := (P_{E} + P_{B}) ∩ Γ_{E} , \\ G_{1} := g_{1} ∩ Γ_{E} , & G_{2} := g_{2} ∩ Γ_{E} . \end{matrix}  (34)

Then we have

δ_{G} = DV (P̃_{A}, P̃_{B}, G_{1}, G_{2}) .  (35)

We obtain the point P̃_{B} with (10) and (11):

P̃_{B} = [(b_{0} − b_{3}, 2b_{1}, 2b_{2}, b_{3} − b_{0})^{⊤}] .  (36)

The point P̃_{A} is the Stephanos image point of the involution ΠÃ with the fixed points Q_{A} = [(1, i)^{⊤}] and R_{A} = [(1, −i)^{⊤}] :

Ã = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , P̃_{A} = PÃ = [(0, 1, −1, 0)^{⊤}] .  (37)

The points G_{1} and G_{2} are the intersection points of the line P̃_{A} + P̃_{B} and the absolute quadric Q_{42}^{2}. We use the set-up

G_{i} = [α_{i}p̃_{A} + β_{i}p̃_{B}] ; i = 1, 2 .  (38)

The coefficients α_{i} and β_{i} are, on account of (36), (37) and (9), the homogeneous solutions of the quadratic equation

(β_{i}(b_{0} − b_{3}))^{2} − (α_{i} + 2β_{i}b_{1})(α_{i} − 2β_{i}b_{2}) = 0 ; i = 1, 2 .  (39)

Because of (2) the equation (39) is equivalent to

(\frac{α_{i}}{β_{i}} )^{2} + 2(b_{1} − b_{2}) · \frac{α_{i}}{β_{i}} − (λ_{B} − μ_{B})^{2} = 0 ; i = 1, 2 .  (40)

Finally from (35), (38) and (40) follows: ^{3}

δ_{G} = \frac{α_{2}}{β_{2}} : \frac{α_{1}}{β_{1}} = \frac{b_{2} − b_{1} − ν\sqrt{(b_{1} − b_{2})^{2} + (λ_{B} − μ_{B})^{2}}}{b_{2} − b_{1} + ν\sqrt{(b_{1} − b_{2})^{2} + (λ_{B} − μ_{B})^{2}}} ; ν ∈ \{−1, 1\} .  (41)

From (33) and (41) we get the assertion (29).

Case 2: Now let at least two of the fixed points of Π_{A} and Π_{B} - say Q_{A} and R_{A} - be real. Without loss of generality they have the homogeneous coordinates

q_{A} = ( \begin{matrix}1 \\ 0 \end{matrix} ) , r_{A} = ( \begin{matrix}0 \\ 1 \end{matrix} ) .  (42)

It follows that

δ_{F} = \frac{q_{B1}}{q_{B0}} : \frac{r_{B1}}{r_{B0}} .  (43)

The entry b_{1} of the transformation matrix B of Π_{B} is not zero; otherwise the point R_{A} = [(0, 1)^{⊤}] would be a fixed point of Π_{B} in contradiction to the assumption Q_{B} ≠ R_{A} ≠ R_{B}. Therefore we get from (43) by application of (3):

δ_{F} = \frac{λ_{B} − b_{0}}{μ_{B} − b_{0}} .  (44)

The points P̃_{A}, P̃_{B}, G_{1} and G_{2} are defined as in (34). The point P̃_{A} is the Stephanos image point of the involution ΠÃ with the fixed points Q_{A} = [(1, 0)^{⊤}] and R_{A} = [(0, 1)^{⊤}] :

Ã = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , P̃_{A} = PÃ = [(1, 0, 0, −1)^{⊤}] .  (45)

The coefficients α_{i} and β_{i} in (38) are, on account of (36), (45) and (9), the homogeneous solutions of the quadratic equation

(α_{i} + β_{i}(b_{0} − b_{3}))^{2} + 2β_{i}b_{1} · 2β_{i}b_{2} = 0 ; i = 1, 2 .  (46)

3 Changing the sign of the factor ν ∈ {−1, 1} corresponds to the exchange of the two (not distinguishable) lines g_{1} and g_{2}.

Because of (2) the equation (46) is equivalent to

(\frac{α_{i}}{β_{i}} )^{2} + 2(b_{0} − b_{3}) · \frac{α_{i}}{β_{i}} + (λ_{B} − μ_{B})^{2} = 0 ; i = 1, 2 .  (47)

Finally we get:

δ_{G} = \frac{b_{3} − b_{0} − 2ν\sqrt{−b_{1}b_{2}}}{b_{3} − b_{0} + 2ν\sqrt{−b_{1}b_{2}}} ; ν ∈ \{−1, 1\} .  (48)

From (44) and (48) follows again the assertion (29).

Remark R6. If T_{1} and T_{2} are the intersection points of the line P_{A} + P_{B} and the tangent cone Q_{31}^{2} from P_{E} to Q_{42}^{2}, then we have for δ_{G}:

δ_{G} = DV (P_{A}, P_{B}, T_{1}, T_{2}) .  (49)

Remark R7. Since the cross ratios δ_{F} and δ_{G} are different from 1, (29) is equivalent to

(\frac{1 + δ_{F}}{1 − δ_{F}} )^{2} · (\frac{1 + δ_{G}}{1 − δ_{G}} )^{2} = (\frac{1 + δ_{F}}{1 − δ_{F}} )^{2} + (\frac{1 + δ_{G}}{1 − δ_{G}} )^{2} .  (50)

Remark R8. From (50) we obtain the equivalence

δ_{F} = −1 ⟺ δ_{G} = −1 .  (51)

Therefore the four fixed points Q_{A},R_{A},Q_{B} and R_{B} of Π_{A} and Π_{B} are in harmonic position, if and only if the two lines P_{E} + P_{A} and P_{E} + P_{B} are orthogonal. ^{4}

Remark R9. In the case δ_{F} ≠ −1, δ_{G} ≠ −1 (50) has the simpler form:

(\frac{1 − δ_{F}}{1 + δ_{F}} )^{2} + (\frac{1 − δ_{G}}{1 + δ_{G}} )^{2} = 1 .  (52)

Lemma 6. Consider the situation given in Proposition 5. If the conic

Q_{31}^{1}(A, B) := ε_{AB} ∩ Q_{42}^{2}  (53)

is regarded as an absolute conic, the plane ε_{AB} becomes a hyperbolic plane P_{|1}^{2}(A, B) , and the following statement holds: The point P_{E} is a point of the interior domain of Q_{31}^{1}(A, B), if and only if the four fixed points Q_{A}, R_{A}, Q_{B} and R_{B} of Π_{A} and Π_{B} are all real and the points Q_{A} and R_{A} are separated on the projective line P^{1} by Q_{B} and R_{B} (δ_{F} < 0).

4 See [3], p. 72, Satz 1 and pp. 201-205, Abschnitt 9D.

Proof: Since the four fixed points of Π_{A} and Π_{B} are pairwise distinct, the plane ε_{AB} = P_{E} + P_{A} + P_{B} is not a tangent plane of the absolute quadric Q_{42}^{2} (see Proposition 1). Hence the conic Q_{31}^{1}(A, B) is non-degenerate. The point P_{E} lies in the interior domain of Q_{31}^{1}(A, B), if and only if the plane ε_{AB} intersects the tangent cone Q_{31}^{2} from P_{E} to Q_{42}^{2} in two complex generators. According to the equations (41) and (48) for the cross ratio δ_{G}, which were given in the proof of Proposition 5, this is the case, if and only if the fixed points Q_{A} and R_{A} are real (without loss of generality let Q_{A} = [(1, 0)^{⊤}] and R_{A} = [(0, 1)^{⊤}]) and furthermore −b_{1}b_{2} < 0 holds. From b_{1}b_{2} > 0 follows with (2), (3) and (44), that Q_{B} and R_{B} and also the cross ratio δ_{F} are real. For a real δ_{F}, however, we have

−b_{1}b_{2} < 0 ⟺^{(2)} (λ_{B} − b_{0})(μ_{B} − b_{0}) < 0 ⟺^{(44)} δ_{F} < 0 .  (54)

If, the other way round, δ_{F} is real and negative, then we get −b_{1}b_{2} < 0 directly from (54). Hence the assertion is shown in both directions.

Proposition 7. Let Π_{A} and Π_{B} be two directly hyperbolic projectivities of P^{1} with altogether four pairwise distinct fixed points Q_{A}, R_{A}, Q_{B} and R_{B} and the characteristic cross ratios c_{A} and c_{B} (c_{A}, c_{B} > 0). Let the fixed points of Π_{A} be separated on P^{1} by the fixed points of Π_{B}. (Thus the cross ratio δ_{F} = DV (Q_{A}, R_{A}, Q_{B}, R_{B}) is negative.) The connecting plane ε_{AB} of the Stephanos image points P_{A}, P_{B} and P_{E} of Π_{A}, Π_{B} and the identity becomes a hyperbolic plane P_{|1}^{2}(A, B), if the (non-degenerate) conic Q_{31}^{1}(A, B) = ε_{AB} ∩Q_{42}^{2} (see (53)) is regarded as an absolute conic. Then the following statement holds: The three points P_{E}, P_{A} and P_{B} of P_{|1}^{2}(A, B) are points of the interior domain of the absolute conic Q_{31}^{1}(A, B). The angle α =<) (P_{E} + P_{A}, P_{E} + P_{B}) (0 < α ≤ \frac{π}{2} ) and the distance d(P_{A}, P_{B}) are given by

cos α = |\frac{1 + δ_{F}}{1 − δ_{F}} | ,  (55)

cosh d(P_{A}, P_{B}) = \frac{1}{4\sqrt{c_{A}c_{B}}} ((1 + c_{A})(1 + c_{B}) − cos α · (1 − c_{A})(1 − c_{B})) . (56)

Proof: According to Lemma 6, the point P_{E} lies in the interior domain of Q_{31}^{1}(A, B). Because of c_{A} > 0 and c_{B} > 0 neither P_{E} and P_{A} nor P_{E} and P_{B} are separated on their connecting line by the (real) intersection points of P_{E} +P_{A} and P_{E} +P_{B}, respectively, with Q_{42}^{2} (see (19)). Hence the points P_{A} and P_{B} also lie in the interior domain of Q_{31}^{1}(A, B). For the angle α = <) (P_{E} + P_{A}, P_{E} + P_{B}) we get with [3], p. 177, Satz 3 (δ_{G} as in (28)):

α = \frac{1}{2i} ln δ_{G} , δ_{G} = e^{2iα} = (cos^{2} α − sin^{2} α) + i · 2 sin α cos α .  (57)

From (57) and (50) we get

cos^{2} α = \frac{(1 + δ_{G})^{2}}{4δ_{G}} = (\frac{1 + δ_{F}}{1 − δ_{F}} )^{2} .  (58)

Because of the restriction 0 < α ≤ \frac{π}{2} we have cos α ≥ 0, and therefore the equation (55) holds. The first cosine law of hyperbolic trigonometry (see [3], p. 306, Satz 1) yields

cosh d(P_{A}, P_{B}) = cosh d(P_{E}, P_{A}) · cosh d(P_{E}, P_{B})−
− cos α · sinh d(P_{E}, P_{A}) · sinh d(P_{E}, P_{B}) .

(59)

The assertion (56) follows with equation (27).

Proposition 8. Let Π_{A}, Π_{B} and Π_{C} be three projectivities of P^{1}, all being different from the identity, with altogether six (real or complex) fixed points Q_{A}, R_{A}, Q_{B}, R_{B}, Q_{C} and R_{C}. Let δ_{AB}, δ_{AC} and δ_{BC} be the cross ratios

\begin{matrix}δ_{AB} := DV (Q_{A}, R_{A}, Q_{B}, R_{B}) , δ_{AC} := DV (Q_{A}, R_{A}, Q_{C}, R_{C}) , \\ δ_{BC} := DV (Q_{B}, R_{B}, Q_{C}, R_{C}) . \end{matrix}  (60)

Furthermore let P_{A}, P_{B}, P_{C} and P_{E} be the Stephanos image points of Π_{A}, Π_{B}, Π_{C} and the identity, let

ε_{AB} := P_{E} + P_{A} + P_{B} , ε_{AC} := P_{E} + P_{A} + P_{C} ,  (61)

and let τ_{1}^{A} and τ_{2}^{A} be the (real or complex) tangent planes of the absolute quadric Q_{42}^{2} ⊂ P_{|2}^{3} which contain the line P_{E} + P_{A} and δ_{ε} the cross ratio

δ_{ε} := DV (ε_{AB}, ε_{AC}, τ_{1}^{A}, τ_{2}^{A}) .  (62)

Then the following equation holds:

\frac{(1 + δ_{ε})^{2}}{δ_{ε}} = \frac{1}{δ_{AB}δ_{AC}} (\frac{(δ_{AB} + δ_{AC}) − δ_{BC}(1 + δ_{AB}δ_{AC})}{1 − δ_{BC}})^{2} =: f(δ_{AB}, δ_{AC}, δ_{BC}) . (63)

Proof: Without loss of generality, each of the fixed points Q_{B} ,R_{B} ,Q_{C} and R_{C} of Π_{B} and Π_{C} is different from the point [(0, 1)^{⊤}]. Then the entries b_{1} and c_{1} of the transformation matrices B and C of Π_{B} and Π_{C} (see (1)) are not zero, and the following quantities can be defined (λ_{B}, μ_{B}, λ_{C} and μ_{C} denote the eigenvalues of B and C):

β_{λ} := \frac{λ_{B} − b_{0}}{b_{1}} , β_{μ} := \frac{μ_{B} − b_{0}}{b_{1}} , γ_{λ} := \frac{λ_{C} − c_{0}}{c_{1}} , γ_{μ} := \frac{μ_{C} − c_{0}}{c_{1}} .  (64)

With (2) we get from (64):

β_{λ}β_{μ} = −\frac{b_{2}}{b_{1}} , γ_{λ}γ_{μ} = −\frac{c_{2}}{c_{1}} , β_{λ} + β_{μ} = \frac{b_{3} − b_{0}}{b_{1}} , γ_{λ} + γ_{μ} = \frac{c_{3} − c_{0}}{c_{1}} .  (65)

Because of (3) we have for the cross ratio δ_{BC}:

δ_{BC} = \frac{γ_{λ} − β_{λ}}{γ_{λ} − β_{μ}} : \frac{γ_{μ} − β_{λ}}{γ_{μ} − β_{μ}} .  (66)

Case 1: Let the fixed points Q_{A} and R_{A} of Π_{A} be real. Then without loss of generality they have the homogeneous coordinates q_{A} = (1, 0)^{⊤} and r_{A} = (0, 1)^{⊤}, and from (44) and (64) follows

δ_{AB} = \frac{β_{λ}}{β_{μ}} , δ_{AC} = \frac{γ_{λ}}{γ_{μ}} .  (67)

From (66) and (67) we get: ^{5}

f(δ_{AB}, δ_{AC}, δ_{BC}) = \frac{(β_{λ}β_{μ} + γ_{λ}γ_{μ})^{2}}{β_{λ}β_{μ} · γ_{λ}γ_{μ}} =^{(65)} \frac{(b_{1}c_{2} + b_{2}c_{1})^{2}}{b_{1}b_{2} · c_{1}c_{2}} .  (68)

If P̃_{A} denotes the intersection point of the line P_{E} +P_{A} and the polar plane Γ_{E} of P_{E} with respect to Q_{42}^{2}, then the planes ε_{AB} and ε_{AC} defined in (61) can be written as:

ε_{AB} = P_{E} + P̃_{A} + P_{B} , ε_{AC} = P_{E} + P̃_{A} + P_{C} .  (69)

With (9), (10) and (45) we obtain the homogeneous coordinates of the poles H_{AB} and H_{AC} of ε_{AB} and ε_{AC} with respect to Q_{42}^{2}:

h_{AB} = (0, b_{1}, −b_{2}, 0)^{⊤} , h_{AC} = (0, c_{1}, −c_{2}, 0)^{⊤} .  (70)

If H_{1} and H_{2} denote the intersection points of the line H_{AB} + H_{AC} and Q_{42}^{2}, then we have

δ_{ε} = DV (H_{AB}, H_{AC}, H_{1}, H_{2}) .  (71)

By using the set-up

H_{i} = [σ_{i}h_{AB} + τ_{i}h_{AC}] ; i = 1, 2 ,  (72)

we get the coefficients σ_{i} and τ_{i} because of (9) and (70) as homogeneous solutions of the quadratic equation

(σ_{i}b_{1} + τ_{i}c_{1})(σ_{i}b_{2} + τ_{i}c_{2}) = 0 .  (73)

Hence follows

δ_{ε} = \frac{σ_{2}}{τ_{2}} : \frac{σ_{1}}{τ_{1}} ,  (74)

and therefore

\frac{(1 + δ_{ε})^{2}}{δ_{ε}} = \frac{(σ_{1}τ_{2} + σ_{2}τ_{1})^{2}}{σ_{1}σ_{2} · τ_{1}τ_{2}} =^{(73)} \frac{(b_{1}c_{2} + b_{2}c_{1})^{2}}{b_{1}b_{2} · c_{1}c_{2}} .  (75)

According to (68) and (75) the assertion (63) holds.

Case 2: Let the fixed points Q_{A} and R_{A} of Π_{A} be conjugate complex. Without loss of generality their homogeneous coordinates are q_{A} = (1, i)^{⊤} and r_{A} = (1, −i)^{⊤}. With (31) and (64) follows:

δ_{AB} = \frac{β_{λ} − i}{β_{λ} + i} : \frac{β_{μ} − i}{β_{μ} + i} , δ_{AC} = \frac{γ_{λ} − i}{γ_{λ} + i} : \frac{γ_{μ} − i}{γ_{μ} + i} .  (76)

5 Since Q_{B}, R_{B}, Q_{C} and R_{C} are different from Q_{A} = [(1, 0)^{⊤}] according to the supposition, b_{2} and c_{2} are not zero.

From (66) and (76) we get:

f(δ_{AB}, δ_{AC}, δ_{BC}) = 4 · \begin{matrix}((1 − β_{λ}β_{μ})(1 − γ_{λ}γ_{μ}) + (β_{λ} + β_{μ})(γ_{λ} + γ_{μ})) & 2 \\ ((1 − β_{λ}β_{μ})^{2} + (β_{λ} + β_{μ})^{2}) & ((1 − γ_{λ}γ_{μ})^{2} + (γ_{λ} + γ_{μ})^{2}) \end{matrix} . (77)

With (9), (10) and (37) we obtain the homogeneous coordinates of the poles H_{AB} and H_{AC}:

\begin{matrix}h_{AB} = (b_{1} + b_{2}, b_{3} − b_{0}, b_{3} − b_{0}, −(b_{1} + b_{2}))^{⊤} , \\ h_{AC} = (c_{1} + c_{2}, c_{3} − c_{0}, c_{3} − c_{0}, −(c_{1} + c_{2}))^{⊤} . \end{matrix}  (78)

Because of (9) and (78) we get the coefficients σ_{i} and τ_{i} in (72) as homogeneous solutions of the quadratic equation

(σ_{i}(b_{1} + b_{2}) + τ_{i}(c_{1} + c_{2}))^{2} + (σ_{i}(b_{3} − b_{0}) + τ_{i}(c_{3} − c_{0}))^{2} = 0 ; i = 1, 2 .  (79)

From (74) and (79) follows:

\frac{(1 + δ_{ε})^{2}}{δ_{ε}} = 4 · \begin{matrix}((b_{1} + b_{2})(c_{1} + c_{2}) + (b_{3} − b_{0})(c_{3} − c_{0})) & 2 \\ ((b_{1} + b_{2})^{2} + (b_{3} − b_{0})^{2}) & ((c_{1} + c_{2})^{2} + (c_{3} − c_{0})^{2}) \end{matrix} .  (80)

According to (65), (77) and (80) the assertion (63) holds.

Remark R10. The planes ε_{AB} and ε_{AC} are identical, if and only if δ_{ε} = 1. With (63) follows:

P_{E}, P_{A}, P_{B}, P_{C} coplanar ⟺ (δ_{AB}δ_{AC}δ_{BC} − (δ_{AB} + δ_{AC} + δ_{BC}) + 1)^{2} =

2δ_{AB}δ_{AC}δ_{BC} + (1 − 2δ_{AB})(1 − 2δ_{AC})(1 − 2δ_{BC}) . (81)

Remark R11. According to equation (63) the planes ε_{AB} and ε_{AC} are orthogonal (δ_{ε} = −1), if and only if δ_{AB} + δ_{AC} = (1 + δ_{AB}δ_{AC}) · δ_{BC}. Since the cross ratio of four pairwise distinct points is different from 1, we have 1 + δ_{AB}δ_{AC} ≠ 0, and therefore

ε_{AB} ⊥ ε_{AC} ⟺ δ_{BC} = \frac{δ_{AB} + δ_{AC}}{1 + δ_{AB}δ_{AC}} .  (82)

Remark R12. If the fixed points Q_{A} and R_{A} of Π_{A} are complex, then the line P_{E} +P_{A} is a passant according to Remark R1, and we get for the angle β :=<) (ε_{AB}, ε_{AC}) (0 ≤ β ≤ \frac{π}{2} ) (see [3], p. 177-178, Satz 3):

β = \frac{1}{2i} · ln δ_{ε} , cos^{2} β = \frac{(1 + δ_{ε})^{2}}{4δ_{ε}} .  (83)

If the fixed points Q_{A} and R_{A} are real, then P_{E} + P_{A} is a secant. If furthermore the planes ε_{AB} and ε_{AC} are not separated by the (in this case real) tangent planes τ_{1}^{A} and τ_{2}^{A} (δ_{ε} > 0), then we have for the angle β (β ≥ 0):

β = \frac{1}{2} · ln δ_{ε} , cosh^{2} β = \frac{(1 + δ_{ε})^{2}}{4δ_{ε}} .  (84)

From (83), (84) and (63) we get a formula for cos β and cosh β, respectively, depending on δ_{AB}, δ_{AC} and δ_{BC}.

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Received November 21, 1994.