3Webs generated by confocal conics and circles
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Abstract
We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3webs and find the exact formulas describing them.
Keywords
3Webs Confocal conics Apollonian circlesMathematics Subject Classification
53A601 Introduction
The concept of webs was invented by Blaschke [3] and connected with many parts of Geometry. Let us recall, that a trivial 3web in a planar domain \(\Omega \) are three families of smooth curves such that there is a diffeomorphism \(\varphi :\Omega \rightarrow \Omega ' \subset \mathbb {R}^2\) taking the families to sets of lines parallel to the sides of a fixed triangle. See [7, Lecture 18] as an introduction to the topic. There are several nontrivial 3webs formed by line and circlefamilies, see [10] for history of the problem and new examples of such webs.
In this paper we consider webs in the positive quadrant \(\mathbb {R}^2_+\) formed by confocal conics and pencils of Apollonian circles, which are defined in the following way: Let \(F_1\) and \(F_2\) be two points in the plane called foci. The family of circles passing through \(F_1\) and \(F_2\) is called an elliptic Apollonian pencil. The pencil of circles orthogonal to all the circles of the first family is called a hyperbolic Apollonian pencil. Each circle from the latter family is a locus of points X such that \(XF_1/XF_2=\mathop {\mathrm {const}}\). We refer to [2, 8] for this and other classical results related with circles and conics.
 (1)

Both families of Apollonian circles and confocal hyperbolas with foci \(F_1\) and \(F_2\);
 (2)

Both families of Apollonian circles and confocal ellipses with foci \(F_1\) and \(F_2\);
 (3)

Families of confocal ellipses, confocal hyperbolas and the hyperbolic Appolonian pencil with foci \(F_1\) and \(F_2\);
 (4)

Families of confocal ellipses, confocal hyperbolas and the elliptic Appolonian pencil with foci \(F_1\) and \(F_2\).
We prove that these webs are trivial by showing a diffeomorphism from a domain \(\Omega \subset \mathbb {R}^2\) to the positive quadrant \(\mathbb {R}^2_+\) which maps horizontal, vertical and “diagonal” (\(x+y\) or \(xy\) is a constant) lines to considered curves. For webs 3 and 4 the images of both diagonal direction are remarkable curve: Apollonian circles and lines (vertical or horizontal).
Before going to the proof, let us say how these pencils relate with each other in an algebraic sense. All circles in the plane can be considered as conics passing through two fixed points of the complex infinite line. These points are called circular and have homogeneous coordinates \(I_1=(1,i,0)\) and \(I_2=(1,i,0)\).
Böhm in [4] constructed a net consisting of lines touching a conic. Quadrilaterals formed by lines of this net can be circumscribed around circles and points of intersection of these lines can be split into families lying on confocal conics. This construction was rediscovered and generalized by the author and Bobenko in [1], where also it was noticed that Böhm’s net is a special case of the Poncelet grid introduced and investigated by Schwarz [11], see also [9] for an additional discussion.
The current constructions were inspired by the work of Edelsbrunner [5, 6], where he invented a new approach for designing smooth surfaces from a set of spheres. His method of connecting spheres by circumscribed hyperboloids is based on the following observation. Let \(k_1\), \(k_2\) satisfy \(k_1^2+k_2^2=1\). If we scale each elliptic Apollonian circle of points \(F_1\) and \(F_2\) \(k_1\) times and each hyperbolic Apollonian circle \(k_2\) with respect to its center, the obtained circles touch a fixed hyperbola with foci \(F_1\) and \(F_2\). Figure 3 shows the case \(k_1=k_2=\frac{1}{\sqrt{2}}\), when the hyperbola is equilateral. Note that if \(k_1>1\), then scaled elliptic Apollonian circles touch an ellipse with foci \(F_1\) and \(F_2\) (Fig. 4). (The touching points may have complex coordinates.)

\(f(P)=\displaystyle \frac{a(P)}{b(P)}\). The locus of points \(f(P)=\mathop {\mathrm {const}}\), is a hyperbolic Apollonian circle corresponding to the points \(F_1\) and \(F_2\). For points \(P\in \mathbb {R}^2_+\) we have \(f(P)>1\).

\(g(P)=\displaystyle \frac{a(P)^2+b(P)^24}{2a(P)b(P)}\). The locus of points \(g(P)=\mathop {\mathrm {const}}\in (0, 1)\), \(P\in \mathbb {R}^2_+\), is the upper arc of a circle passing through \(F_1\) and \(F_2\) (and the arc symmetric to it in xaxis). Indeed, for any point P on this arc we have \(\cos \angle F_1PF_2=g(P)=\mathop {\mathrm {const}}\).

\(h(P)=a(P)b(P)\). The locus of points \(h(P)=\mathop {\mathrm {const}}\) is a branch of a hyperbola with foci \(F_1\) and \(F_2\). Note that for \(P \in \mathbb {R}^2_+\) we have \(0<h(P)<2\).

\(e(P)=a(P)+b(P)\). The locus of points \(e(P)=\mathop {\mathrm {const}}\) is an ellipse with foci \(F_1\) and \(F_2\).
2 The web from Apollonian circles and confocal hyperbolas
In this section we prove that two pencils of Apollonian circles with foci at \(F_1\) and \(F_2\), and the family of hyperbolas with foci at \(F_1\) and \(F_2\) form a trivial web in the positive quadrant \(\mathbb {R}_+^2\). Applying the map to vertices of a shifted lattice \(k\mathbb {Z}^2+t\) we obtain a configuration shown on Fig. 5.
Theorem 1
Proof
3 The web from Apollonian circles and confocal ellipses
We prove that two pencils of Apollonian circles with foci at \(F_1\) and \(F_2\), and the family of ellipses with foci at \(F_1\) and \(F_2\) form a trivial web in the positive quadrant \(\mathbb {R}_+^2\) (Fig. 6).
Theorem 2
Proof
4 The web from confocal conics and a hyperbolic Apollonian pencil
In this section we prove that there is a trivial web formed by confocal ellipses and hyperbolas, and the hyperbolic Apollonian pencil. The family of vertical lines can be adjoint to this web in a very natural way (Fig. 7).
Theorem 3
Proof
5 The web from confocal conics and an elliptic Apollonian pencil
The last example of a trivial web is formed by confocal ellipses and hyperbolas, and and elliptic Apollonian pencil, and family of horisontal lines, which can be adjoint to the web as well (Fig. 8).
Theorem 4
Proof
Notes
Acknowledgements
Open access funding provided by Institute of Science and Technology (IST Austria).
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