6 Connecting quadrics and cycles

For introduction of the fundamental concept of cycle in the context of MoebInv library see the Jupyter notebook What is a cycle?

6.1 The point space and the cycle space

Let us start from the familiar case of circles. The graphical view represents a two-dimensional plane—the collection of points with coordinates (x,y). We will call it the point space.

A circle with centre (x₀,y₀) and radius r is described by equations

It can be equivalently written as:

Thus we will use the four real numbers (k,l,n,m) to represent a cycle

Clearly coefficients (k,l,n,m) and (λ k, λ l, λ n, λ m) with a non-zero real λ represent the same cycle and will be considered equivalent. Thus we can view a cycle either as the geometric set of points (u,v) satisfying the equation (4) or equivalent tuples (λ k, λ l, λ n, λ m), λ ≠ 0 of coefficients. The former viewpoint was already called point space and the later will be labelled as the cycle space.

6.2 Elliptic, parabolic, hyperbolic metrics and cycle product

It is interesting to make a generalisation by defining a cycle by the equation

That is for the three values of the parameter σ we obtain equations of circles, parabolas and hyperbolas. Therefore, these three situations are called elliptic, parabolic, and hyperbolic metrics in the point space.

It is turn out that the cycle space can be also meaningfully equipped with metric of these three flavours. There is an cycle product of cycles C₁ and C₂ with the explicit expression

It has several important properties:

1. It is symmetric: ⟨C₁,′ ₂⟩ = ⟨C₂,′ ₁⟩.
2. It is linear in coefficients of the cycle C₁ as well as C₂.
3. Its is invariant under the fractional linear transforms F, that is: ⟨FC₁,F′ ₂⟩_σ = ⟨C₁,′ ₂⟩_σ if σ=σ.
4. If the fractional linear transforms F∈SL₂(ℝ) (that is, has only real coefficients and is Möbius transform) then the product (6) is F-invariant for all nine independent combinations of σ and σ [3, § 5.2].

The three values σ =−1, 0, or 1 in the cycle product (6) are called elliptic, parabolic and hyperbolic metric in the cycle space, respectively. The Yaglom allows to switch to combination of metrics in point and cycle spaces at any time.

6.3 From space of cycle to space of points and return back

We need a vocabulary, which translates geometric properties of cycles on the point space to corresponding relations in the cycle space. The key ingredient is the cycle product (6). As usual, the relation ⟨C₁,′ ₂ ⟩=0 is called the orthogonality of cycles C₁ and C₂. If metrics of point and cycles spaces agree, the orthogonality corresponds to orthogonality of cycles in the point space in the respective metrics.

As was pointed out above the orthogonality is a liner condition on the cycles’ coefficients. However, certain other relations, e.g. tangency of cycles, involve polynomials of cycle products and thus are non-linear. For a successful algorithmic implementation, the following observation is important: all non-linear conditions below can be linearised if the additional quadratic condition of normalisation type is imposed:

Here we provide connections of the cycle product with other cycle relations discussed in § 2.4.

1. A quadric is a line (i.e. is a hyperplane), that is, its equation is linear. Then, either of two equivalent conditions can be used:
   1. k component of the cycle vector is zero;
   2. the cycle is orthogonal ⟨C₁,′ _∞ ⟩=0 to the “zero-radius cycle at infinity” ′ _∞=(0,0,1).
2. A quadric on the plane represents a line in Lobachevsky geometry if it is orthogonal ⟨C₁,′ _ℝ ⟩=0 to the real line cycle C_ℝ. A similar condition is meaningful in higher dimensions as well.
3. A quadric C represents a point, that is, it has zero radius at given metric of the point space, if the cycle is self-orthogonal (isotropic): ⟨C,′ ⟩=0. Naturally, such a cycle cannot be normalised to the form (7).
4. Two cycles are orthogonal in the point space. Then the cycles are orthogonal in the sense of the inner product (6).
5. Two cycles C and ′ are tangent. Then we have the following quadratic condition:

   ⟨ ⟨

   C,G

   ⟩ ⟩

   2 =

   ⟨ ⟨

   C,C

   ⟩ ⟩

   ⟨ ⟨

   G ,G

   ⟩ ⟩

   ⎛ ⎝

   or

   ⎡ ⎣

   C,G

   ⎤ ⎦

   =± 1

   ⎞ ⎠

   . (8)

   With the assumption, that the cycle C is normalised by the condition (7), we may re-state this condition in the relation, which is linear to components of the cycle C:

   ⟨ ⟨

   C,G

   ⟩ ⟩

   = ±

   √

   ⟨ ⟨ G ,G   ⟩ ⟩

   . (9)

   Different signs here represent internal and outer touch.
6. Inversive distance θ of two (non-isotropic) cycles is defined by the formula:

   ⟨ ⟨

   C,G

   ⟩ ⟩

   = θ

   √

   ⟨ ⟨ C,C  ⟩ ⟩

   √

   ⟨ ⟨ G ,G   ⟩ ⟩

   (10)

   In particular, the above discussed orthogonality corresponds to θ=0 and the tangency to θ=±1. For intersecting spheres θ provides the cosine of the intersecting angle. For other metrics, the geometric interpretation of inversive distance shall be modified accordingly.

   If we are looking for a cycle C with a given inversive distance θ to a given cycle G , then the normalisation (7) again turns the defining relation (10) into a linear with respect to parameters of the unknown cycle C.

7. A generalisation of Steiner power d of two cycles is defined as, cf. [1, § 1.1]:

   d=

   ⟨ ⟨

   C,G

   ⟩ ⟩

   +

   √

   ⟨ ⟨ C,C  ⟩ ⟩

   √

   ⟨ ⟨ G ,G   ⟩ ⟩

   , (11)

   where both cycles C and G are k-normalised, that is the coefficient in front the quadratic term in (5) is k=1. Geometrically, the generalised Steiner power for spheres provides the square of tangential distance. However, this relation is again non-linear for the cycle C.

   If we replace C by the cycle C₁=1/√⟨C,C ⟩C satisfying (7), the identity (11) becomes:

   d· k=

   ⟨ ⟨

   C1,G

   ⟩ ⟩

   +

   √

   ⟨ ⟨ G ,G   ⟩ ⟩

   , (12)

   where k=1/√⟨C,C ⟩ is the coefficient in front of the quadratic term of C₁. The last identity is linear in terms of the coefficients of C₁.

8. A cycles cross ratio of four cycles C₁, C₂, C₃ and C₄ is:

   [C1, C2; C3, C4]=

   ⟨ ⟨ C1,C3  ⟩ ⟩ ⟨ ⟨ C1,C4  ⟩ ⟩

   :

   ⟨ ⟨ C2,C3  ⟩ ⟩ ⟨ ⟨ C2,C4  ⟩ ⟩

   (13)

   assuming ⟨C₁,C₄ ⟩ ⟨C₂,C₃ ⟩ ≠ 0. See [10, 9] for further discussion.

Summing up: if an unknown cycle is connected to already given cycles by any combination of the above relations, then all conditions can be expressed as a system of linear equations for coefficients of the unknown cycle and at most one quadratic equation (7).