Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 299-305.

To the Isotropic Analogue of the Theorem of Abramescu

Jürgen Tölke

Mathematisches Institut, Universität Siegen, Hölderlinstraße 3, D-57068 Siegen, Deutschland

MSC 1991: 51N25

Keywords: Isotropic plane, admissible quadrilateral, circular and completely circular curves of third order

Let c be an admissible C^{ω}-curve segment [7, p.108] free of inflection points in an isotropic plane. The tangents t_{1}, t_{2}, t_{3} of the neighboring curve points P_{1}, P_{2}, P_{3} determine a trilateral Δ. H. Sachs [6] proved that the isotropic circle through the vertices of Δ converges under the limiting process P_{2} → P_{1}, P_{3} → P_{1} to a limit circle, which is called the isotropic Abramescu circle K_{A}(P_{1}) of c at P_{1}. In analogy we will consider the quadrilateral of the tangents t_{1}, …, t_{4} of four neighboring points P_{1}, P_{2}, P_{3}, P_{4} of c and the completely circular curve of third order Γ [4, 9] through the six intersection points of the tangents t_{1}, t_{2}, t_{3}, t_{4}. We will prove that Γ converges under the limiting process P_{2} → P_{1}, P_{3} → P_{1}, P_{4} → P_{1} to a limit cubic Γ^{∗}. For non-vanishing second isotropic curvature of c at P_{1}, the cubic Γ^{∗} is a divergent cubical parabola which has P_{1} as a doublepoint. The cubic Γ^{∗} touches the affinparabola of c in P_{1} in the characteristic points of the Abramescu circle K_{A}(P_{1}).

1. In analogy to an admissible trilateral [7, p.22] we call a quadrilateral (t_{1}, t_{2}, t_{3}, t_{4}) of an isotropic plane I_{2}(ℝ) admissible, if this quadrilateral is complete in the projective plane P_{2}(ℝ) ⊃ I_{2} and if no side t_{1} and no intersection point T_{ik} := t_{i} ∧ t_{k} = T_{ki} is incident with the absolute point F and the absolute straight line f respectively. Under the general isotropic similarity group of I_{2} we have two classes of admissible quadrilaterals: The circle quadrilaterals and the parabola quadrilaterals according to wether the quadrilateral (t_{1}, t_{2}, t_{3}, t_{4}) is a tangent quadrilateral of a circle or a parabola.

In generalizing the classical notation “circular”, H. Sachs [8] calls an algebraic curve K^{(n)} with the order n of P_{2}(ℝ) circular in I_{2} ⊂ P_{2}(ℝ), if the absolute point F is at least an (n − 1)-fold intersection point of K^{(n)} with the absolute straight line f. In the

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1997 Heldermann Verlag special case that F is an n-fold intersection point of K^{(n)} and f, D. Palman [4] calls K^{(n)} completely circular. So we can regard the circumcircle of an admissible triangle Δ of I_{2} as the completely circular algebraic curve of lowest order which contains the vertices of Δ. In analogy we define the circumcurve U of an admissible quadrilateral (t_{1}, t_{2}, t_{3}, t_{4}) as the completely circular curve of lowest order which contains the points T_{ik}. If the quadrilateral is a circal quadrilateral, the circumcurve U is a (possible reducible) Tridens curve of third order, and if it is a parabola quadrilateral, then U is a divergent Parabola of third order [10,11].

2. We fix a (ξ, η)-coordinate system in such a way that in the affine plane A_{2}(ℝ) ⊃ I_{2} the absolute point lies on the η-axis. By means of an isotropic motion we can assume the (ξ, η)-coordinates of the points T_{13}, T_{14}, T_{34} of the admissible quadrilateral (t_{1}, t_{2}, t_{3}, t_{4}) (A, B, λ ∈ ℝ) are given as follows:

(1)  T_{13} = (0, 0), T_{14} = (A, 0), T_{34} = (λB, B) with λ(λB − A)AB ≠ 0.

It follows

(2) T_{12} = (a, 0), T_{23} = (λb, b), where (a−A)(b−B)(aB−bA)(λb−a)ab ≠ 0, and a, b ∈ ℝ.

For the coordinates of T_{24} =: (ξ_{t}, η_{t}) we have

(3) (aB − bA)ξ_{t} = β, (aB − bA)η_{t} = bB(a − A), β := −aA(b − B) + λbB(a − A).

We obtain for the isotropic line coordinates w : u : v (wη = uξ + v) of the straight lines t_{i} the following values:

(4)  \begin{matrix}t_{1}: & u = v = 0, & t_{2}: & w = a − λb, & u = −b, & v = ab \\ t_{3}: & w = λ, u = 1, v = 0, & t_{4}: & w = A − λB, & u = −B, & v = AB \end{matrix}

Let K(Δ_{i}) be the circumcircle of the admissible trilateral Δ_{i}(t_{j}t_{k}t_{l}), {i, j, k, l}={1, 2, 3, 4}, formed by three different sides t_{j}, t_{k}, t_{l} of an admissible quadrilateral (t_{1}, …, t_{4}). For the pencil K^{(3)}(μ) of the circular curves of third order which contain the points T_{ik} we have

(5)  (aB − bA)K^{(3)}(μ) ≡ (ξ − λη)K(Δ_{3}) + μηK(Δ_{1}) = 0.

Since

\begin{matrix}K(Δ_{1}) ≡ & (aB − bA)(ξ − λb)(ξ − λB) + (b − B)(a − λb)(A − λB)(ξ − λη) = 0 \\ K(Δ_{3}) ≡ & (aB − bA)(ξ − a)(ξ − A) − (a − A)(a − λb)(A − λB)η = 0, \end{matrix}

we find

(6)  \begin{aligned}(aB − bA)τ(μ) := & μ\{λ^{2}bB(b − B) − 2λbB(a − A) + aA(b − B)\} \\ & − λ^{2}bB(a − A) + 2λaA(b − B) − aA(a − A) \end{aligned}

and

(7)  γ(μ) := 2λ(a + A) + τ(μ), δ(μ) := λ{a + A + μ(b + B)} + τ(μ)

for the representation of the pencil K^{(3)}(μ)

(8) K^{(3)}(μ) ≡ ξ(ξ − a)(ξ − A) + (μ − λ)ξ^{2}η + γ(μ)ξη − λδ(μ)η^{2} + λ(λμbB − aA)η = 0,

where τ, γ, δ are defined by (6) and (7). Hence K^{(3)}(λ) is the representation of the circumcurve U of the admissible quadrilateral (t_{1}, t_{2}, t_{3}, t_{4}). Using the abbreviations (6) and (7), we get

(9)  \begin{matrix}(aB − bA)δ(λ) = (a − λb)(λB − A)σ with σ := a − A − λ(b − B) \\ (b − B)(γ(λ) − δ(λ)) = λ[(b + B)σ − 2(aB − bA)], \end{matrix}

and by this, finally,

(10)  K^{(3)}(λ) ≡ ξ(ξ − a)(ξ − A) + γ(λ)ξη − λδ(λ)η^{2} + λ(λ^{2}bB − aA)η = 0.

3. The above preparations enable us to treat the question posed in the introduction. The answer may be regarded as a further generalization [3] of the isotropic theorem of Abramescu [6].

Theorem. Denote by c an admissible C^{ω}-curve and by P_{1} a non-inflection point of c with tangent t_{1}. Let P_{2}, P_{3}, P_{4} be mutually different points of c, which are also distinct from P_{1} and t_{2}, t_{3}, t_{4} their tangents. The completely circular algebraic curve of third order Γ defined by the six tangent intersection points converges under the limiting process P_{2} → P_{1}, P_{3} → P_{1}, P_{4} → P_{1} to a limit curve Γ^{∗} for which the following holds:

(i) If the second isotropic curvature of c vanishes at P_{1}, then the limit-curve Γ^{∗} is a reducible curve of third order. Further Γ^{∗} splits into the isotropic normal through P_{1} and a circle Γ_{k} which touches c at P_{1}. The isotropic curvature of c at P_{1} is the radius of Γ_{k}.

(ii) If the second isotropic curvature of c does not vanish at P_{1}, then the limit-curve Γ^{∗} is a divergent parabola of third order which has P_{1} as a double point. The harmonic polar line of the absolute point (with respect to Γ^{∗}) is the bisectrix of the double point tangents. Further t_{1} is a double point tangent. The limit-cubic Γ^{∗} touches the affinparabola of c in P_{1} in the characteristic points of the Abramescu circle K_{A}(P_{1}) of c at P_{1}.

Proof. (a) The analytical determination of Γ. We choose the isotropic arc length ξ̂ of c as our parameter and the coordinate system (ξ̂, η) in such a way that P_{1} coincides with the origin and the ξ̂-axis with the tangent t_{1}:

(11)  η = f(ξ̂), f(0) = f′(0) = 0, f′′(0) ≠ 0.

By the assumptions on the curve c, the tangents t_{1}, …, t_{4} determine an admissible quadrilateral. Their equations are

(12) t_{1}: η = 0, t_{i}: η_{i} = u_{i}ξ̂+ v_{i}, i = 2, 3, 4 with u_{2}u_{3}u_{4}(u_{2} − u_{3})(u_{2} − u_{4})(u_{3} − u_{4}) ≠ 0.

To determine the completely circular curve of third order Γ which contains the intersection points t_{j} ∧ t_{k}, we translate the origin of the coordinate system so that

(13)  ξ = ξ̂+ \frac{v_{3}}{u_{3}} .

Defining

ρ_{ik} := u_{i}v_{k} − u_{k}v_{i}

we get by means of section 2 the relations

a = \frac{ρ_{23}}{u_{2}u_{3}} , b = \frac{ρ_{23}}{u_{2} − u_{3}} , A = \frac{ρ_{43}}{u_{4}u_{3}} , B = \frac{ρ_{43}}{u_{4} − u_{3}} , λ = \frac{1}{u_{3}} .

It follows that

\begin{matrix}a − λb = \frac{−ρ_{23}}{u_{2}(u_{2} − u_{3})}, A − λB = \frac{−ρ_{43}}{u_{4}(u_{4} − u_{3})}, aB − bA = \frac{(u_{2} − u_{4})ρ_{23}ρ_{43}}{u_{2}u_{4}(u_{2} − u_{3})(u_{4} − u_{3})}, \\ σ = \frac{u_{2}(u_{2} − u_{3})ρ_{43} − u_{4}(u_{4} − u_{3})ρ_{23}}{u_{2}u_{4}(u_{2} − u_{3})(u_{4} − u_{3})} , λ^{2}bB − aA = \frac{(u_{2} − u_{3} + u_{4})ρ_{23}ρ_{43}}{u_{2}u_{3}u_{4}(u_{2} − u_{3})(u_{4} − u_{3})}. \end{matrix}

Using (9) we find that

λδ(λ) = \frac{−σ}{u_{3}(u_{2} − u_{4})}, γ(λ) = δ(λ) − \frac{u_{2}(u_{2} − u_{3})ρ_{43} + u_{4}(u_{4} − u_{3})ρ_{23}}{u_{2}u_{3}u_{4}(u_{2} − u_{3})(u_{4} − u_{3})} .

Hence from (10) and (13) we finally get for the representation of Γ

(14) (ξ̂+ \frac{v_{2}}{u_{2}} )(ξ̂+ \frac{v_{3}}{u_{3}} )(ξ̂+ \frac{v_{4}}{u_{4}} )+γ(λ)ξ̂η−λδ(λ)η^{2}+[\frac{v_{3}}{u_{3}} γ(λ) + λ(λ^{2}bB − aA)]η = 0.

(b) The limitcurve Γ^{∗} of Γ. If the points P_{i} have (ξ̂, η)-coordinates P_{1}(0, 0), P_{i}(h_{i}, f(h_{i})) (i = 2, 3, 4) with h_{2}h_{3}h_{4}(h_{2} − h_{3})(h_{2} − h_{4})(h_{3} − h_{4}) ≠ 0, we get

u_{i} = f′(h_{i}), v_{i} = f(h_{i}) − h_{i}f′(h_{i}), i = 2, 3, 4.

From the Taylor expansion [7, p.117] it follows that

u_{i} = ∑_{ρ=2}^{∞} \frac{1}{(ρ − 1)!}h_{i}^{ρ−1}f^{(ρ)}(0), v_{i} = ∑_{ρ=2}^{∞} \frac{1 − ρ}{ρ!}h_{i}^{ρ}f^{(ρ)}(0).

Writing f^{(ρ)} := f^{(ρ)}(0) we have (i ≠ k)

\begin{matrix}u_{i} − u_{k} = & (h_{i} − h_{k}) \{f′′ + \frac{1}{2}(h_{i} + h_{k})f′′′ + F_{ik}(h_{i}, h_{k})\} \\ u_{i}(u_{i} − u_{k}) = & h_{i}(h_{i} − h_{k}) \{(f′′)^{2} + \frac{1}{2}(h_{k} + 2h_{i})f′′f′′′ + F_{iik}(h_{i}, h_{k})\} \end{matrix}

with

lim_{h_{i},h_{k}→0} F_{ik}(h_{i}, h_{k}) = 0, lim_{h_{i},h_{k}→0} F_{iik}(h_{i}, h_{k}) = 0.

Further it follows that

ρ_{ik} = h_{i}h_{k}(h_{k} − h_{i}) \{−\frac{1}{2}(f′′)^{2} − \frac{1}{3}(h_{i} + h_{k})f′′f′′′ + R_{ik}(h_{i}, h_{k})\} (i ≠ k)

and lim_{h_{i},h_{k}→0} R_{ik}(h_{i}, h_{k}) = 0 [7, p.118]. It follows that

u_{2}(u_{2} − u_{3})ρ_{43} − u_{4}(u_{4} − u_{3})ρ_{23} = h_{2}h_{3}h_{4}(h_{2} − h_{3})(h_{2} − h_{4})(h_{4} − h_{3}) \{\frac{1}{6}(f′′)^{3}f′′′ + S\}

with lim_{h_{2},h_{3},h_{4}→0} S = 0 [7, p.118]. Hence

(15)

lim_{h_{2},h_{3},h_{4}→0} λδ(λ) = −\frac{1}{6} \frac{f′′′(0)}{(f′′(0))^{3}} , lim_{h_{2},h_{3},h_{4}→0} γ(λ) = −f′′(0)^{−1},

lim_{h_{2},h_{3},h_{4}→0} λ (λ^{2}bB − aA) = 0

This means that the completely circular curve Γ of third order determined by the intersection points t_{i} ∧ t_{k} converges to a limit-curve Γ^{∗} with the equation

(16)  ξ̂^{3} − (f′′(0))^{−1}ξ̂η + \frac{1}{6} \frac{f′′′(0)}{(f′′(0))^{3}} η^{2} = 0.

(c) The statements on Γ^{∗}

(i) If the second isotropic curvature κ^{∗}(P_{1}) := f′′′(0) vanishes, (16) splits into the isotropic normal of P_{1} and a circle Γ_{k} which touches c at P_{1}. The radius of Γ_{k} is equal to the curvature κ(P_{1}) = f′′(0) of c. (ii) In case of κ^{∗}(P_{1}) ≠ 0 we apply the isotropic motion

ξ̂ = x, η = 3\frac{(f′′(0))^{2}}{f′′′(0)} x + y.

The curve Γ^{∗} is then in normal form

(17)  x^{2} (x − \frac{3}{2} \frac{f′′(0)}{f′′′(0)}) + \frac{1}{6} \frac{f′′′(0)}{(f′′(0))^{3}} y^{2} = 0.

So Γ^{∗} is a divergent parabola with the double point P_{1}. The tangents at P_{1} are the tangent t_{1} of c and the straight line

(18)  ξ̂− \frac{1}{6} \frac{f′′′(0)}{(f′′(0))^{2}} η = 0.

The bisection of the doublepoint tangents is the harmonic polar line h_{F} [1, p.32] of the absolute point F (with respect to Γ^{∗}). On the other hand t_{1}(P_{1}) is the bisectrix of h_{F} and the affine normal of c at P_{1}

(19)  ξ̂+ \frac{1}{3} \frac{f′′′(0)}{(f′′(0))^{3}} η = 0.

The straight line h_{F} intersects Γ^{∗} at P_{1} and the point H with

(20)  ξ̂_{h} = \frac{3}{2} \frac{f′′(0)}{f′′′(0)}, η_{h} = \frac{9}{2} \frac{f′′(0)^{3}}{(f′′′(0))^{2}} .

The point H is the isotropic focal point of Γ^{∗} and also the characteristic point (different from P_{1}) of K_{A}(P_{1}). So H is the isotropic focal point of the affine parabola of c at P_{1} [5].

Notation:

1... C^{ω}-curve c with the point P_{1} and the curve tangent t_{1}.

2... The circle of Abramescu K_{A}(P_{1}).

3... The special hyperbola [7, p.68] H as the locus of the focal points of the cones which hyperosculate c at P_{1}. The drawn asymptotic lines are parallel to the isotropic direction and to the affine normal of c at P_{1} respectively.

4... The affine parabola of c at P_{1}.

5... The limit cubic Γ^{∗} with the characteristic points P_{1} and H of K_{A}(P_{1}).

Remarks.

1. The isotropic circle Γ_{k} is the curvature circle of Γ^{∗} at the doublepoint P_{1}.

2. There is no Euclidean analogue for Γ^{∗}. O. Röschel [5] studied the isotropic analogon of the Euclidean focal point cubic of A. Cazamian [2, p.264f.]. The locus of the isotropic focal points of all cones which hyperosculate c at P_{1} is a special hyperbola H given by

(21)  η = f′′(0) (ξ̂+ \frac{1}{3} \frac{f′′′(0)}{(f′′(0))^{2}} η) ξ̂.

H touches c at P_{1} and H is a point of H. With (21) we see that the affine normal (19) is the bisectrix of t_{1}(P_{1}), and the diameter h

(22)  ξ̂+ \frac{1}{6} \frac{f′′′(0)}{(f′′(0))^{2}} η = 0

of H through P_{1}. So t_{1}(P_{1}) is the bisectrix of h and a doublepoint tangent (18) of Γ^{∗}.

References

[1] Burau, W.: Algebraische Kurven und Flächen. Bd.I: Algebraische Kurven der Ebene. Sammlung Göschen, Bd.435. Berlin 1962.

[2] Cazamian, A.: Note sur la strophoide. Nouv. Ann. (3) 13 (1894), 264–265.

[3] Kurnik, Z.: Eine Verallgemeinerung des Satzes von N. Abramescu für Kurven der isotropen Ebene. Ber. Math.-Stat. Sekt., Forschungszentrum Graz, 241 (1985), 1–6.

[4] Palman, D.: Über zirkuläre Kurven 3. Ordnung der isotropen Ebene. Rad. JAZU [444] 8 (1989),37–46.

[5] Röschel, O.: Bemerkungen zum Satz von Abramescu in der euklidischen und der isotropen Ebene. Archiv d. Math. 42 (1984), 173–177.

[6] Sachs, H.: Ein isotropes Analogon zu einem Satz von Abramescu und einige Grenzwertformeln. Archiv d. Math. 23 (1972), 661–668.

[7] Sachs, H.: Ebene isotrope Geometrie. Vieweg-Verlag, Wiesbaden 1987.

[8] Sachs, H.: Oskulierende und hyperoskulierende Kegelschnittbüschel der isotropen Ebene. Sitzungsber. d. Österr. Ak. d. Wiss. 196 (1987), 337–375.

[9] Sachs, H.: Vollständig zirkuläre Kurven n-ter Ordnung der isotropen Ebene. Studia Sci. Math. Hungar. 24 (1989),377–383.

[10] Tölke, J.: Zur Geometrie der Tridens-Kurven der isotropen Ebene. To appear in Studia Sci. Math. Hungar.

[11] Tölke, J.: Zur isotropen Vierseitgeometrie. In Preparation.

[12] Tölke, J.: Parabelscharen und Grenzkreise in der isotropen Ebene. J. Geom. 56 (1996), 182–189.

Received July 7, 1996