Quick start with cycle relations

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2 Quick start with cycle relations

Here is a brief summary how various geometric properties can be expressed through predefined cycle relations. Mathematical aspects of cycle relations will be discussed in § 6.3,

A new cycle can be added to a figure by specifying a list of its relation to other cycles—either already existing or the new cycle itself. In this way we can avoid vicious loops. Thus the concept of a cycle relation is central to this software and we systematically introduce it here. Lists of relations for some common configurations are presented in § 2.5.

A cycle relation from MoebInv library belongs to one of the following categories:

Characterising a geometric property through one or several equations on new cycle’s coefficients—it is typically used in a list with others relations. The most fundamental example is orthogonality (⊥), cf. § 2.1.
Constructing a new cycle—it is used alone or with choosing relations from the next category. An example is a reflection defined by a pair “object-mirror” (⟃).
Choosing a cycle by certain condition, e.g. either its all coefficients are real numbers. Such relations are usually not used themselves, rather they may complement relations of the first two categories.

Thereafter, a meaningful list of relations to define a new cycle has the form:

(some characterising or single constructing) and optional choosing relation(s).

The order of relations in the list is not important. Some relations (at any category) may need additional parameter(s) (real numbers).

All relations except four choosing—Only Real coefficients (ℜ), Only Numeric coefficients (½), Positive orientation (⇧) and Negative orientation (⇩) —link a new cycle to other cycle(s) and can be selected from the respective cycle’s context menu (LeftClick).

Besides the above four exclusions the following relations of a cycle to itself are meaningful:

Orthogonal (⊥);
f-Orthogonal (⋋);
Two cycle product ratio (⋇);
Product sign (±);
Positive orientation (⇧) and Negative orientation (⇩) .

For this reason all these relations appear in New cycle ↷ ToolBar button, to give an opportunity to define such relations of the new cycle to itself.

2.1 Characterising relations for common geometric properties

The most used relation in this category is Orhogonality (⊥)—a wealth of geometric properties can be expressed by it:

Geometric property	Cycle relation	Is linear?
Cycle L is a line	L is Orthogonal (⊥) to infinity (infty=∞)	✔
Cycle L is a Lobachevsky line	L is Orthogonal (⊥) to the real line (R=ℝ)	✔
Cycle A is a point	A is Orthogonal (⊥) to itself	✘
Cycle C passes point A	C is Orthogonal (⊥) to A	✔
Line L₁ is parallel to line L₂	L₁ is Tangent (⩈) to L₂	✘

Usually three independent conditions fully determine a cycle on a plane, e.g.:

Geometric object	List of relations
Line L through two different points A, B	{L ⊥ ∞, L ⊥ A, L ⊥B}
Intersection point P of two cycles C and D	{P ⊥ P, P ⊥ C, P ⊥ D}
Cycle C has centre A and passes B	{C ♓{A,∞}, C ⊥ B}

In the first case all equations are linear, thus there is at most one solution. In the second case one condition (P ⊥ P) is quadratic, thus two different solutions may appear. In the third case the relation “in pencil” (♓) spanned by A and ∞ says that C is concentric with A, that is has its centre in A if A is a point. The relation leaves only one degree of freedom, thus the second relation (C ⊥ B) already defines C uniquely.

Other relations in this category are:

Focal orthogonal (⋋);
Tangent both (⩈) subdivided into Tangent inner (⋐) and Tangent outer (⫓);
Steiner power (Ⓢ);
Cross tangent distance (⦻);
ArcCos of cycle angle (∠);
Two cycle product ratio (⋇);
Cycles cross ratio (∺),
In pencil (♓).

They are described below.

2.2 Constructing relations

Some relations can directly construct a new cycle from already defined cycles. Relations in this category are:

The cycle Conjugation of C₁ and C₂ which geometrically means inversion (reflection) of the object C₁ in the mirror C₂. The “mirror” cycle C₂ can be either ad-hoc given by its coefficients (Ad-hoc cycle reflection (⊚) ) or an existing cycle named by its label (Named cycle conjugation (⟃))
The Fraction-linear transform (⧉) of an existing cycle defined by a given complex 2× 2 matrix.
The SL₂(ℝ) transform (⊞) of an existing cycle with a given real 2× 2 matrix,
The selection among multiple instances of an existing cycle the instance Instance with index (#). (Multiple instances appear from quadratic relations, e.g. cycle intersections).

2.3 Choosing relations

We often need to restrict the set of solutions by some additional requirements. For example, the list

{P ⊥ P, P ⊥ L₁, P ⊥ L₂, P≠∞}

determines the unique (if exists) intersection point of two lines L₁ and L₂ in the finite plane. The last condition P≠∞ (that is P is Different from ∞) is required to exclude infinity, which is the common point of any two lines.

Relations in this category are:

A new cycle shall be Different (≠) from an already existing one;
A variant of the previous relations suitable for float-point calculations (or rather complicated expressions with irrationalities) is Almost different (≄).
A new cycle shall have Only real coefficients (ℜ).
A new cycle shall have Only numeric coefficients (½) (without symbolic variables).
A new cycle shall have a definite Product sign (±) in the product with an existing cycle.
A new cycle shall have a definite orientation: Positive orientation (⇧) or Negative orientation (⇩).

2.4 List of all known relations

Besides the three categories, relations have other important properties:

They link a new cycle to various numbers of other cycles (currently from 0 to 3).
They may or may not require additional parameter(s).
They may or may not meaningfully link a new cycle to itself.
They may or may not depend on the metric: elliptic, parabolic and hyperbolic.

In addition to these characterising relations

produce some equations, which may or may not be linear.

The possible linearity is an important property greatly simplifying their solutions. However, none of characterising relation is linear if it links a the new cycle to itself.

We summarise all above-mentioned properties of known relations in reference tables below. The columns are:

The respective encoding character.
Par: The number of required parameters.
Cycles: the number of other cycles involved. If it is meaningful for a relations to link the new cycle to itself, then this is indicated by the symbol ↶in the column with the number of other cycles required. Such relations appear in New cycle ↷ ToolBar button.
Lin? is the relation linear.?
Metric? is the relation metric dependant?
Name of the relation.

It is expected that GUI handles all the above aspects of cycle relations in a correct manner.

2.4.1 Characterising relations properties

The next table shows all characterising relations with

	Par	Cycles	Lin?	Metric?	Name
⊥	0	+1 ↶	✔	✅	Orthogonal
⋋	0	+1 ↶	✔	✅	f-Orthogonal
♓	0	+2	✔		In pencil
⋇	1	+2 ↶	✔^(*)	✅	Two cycle product ratio
∺	1	+3 ↶	✔^(*)	✅	Cycles cross ratio
⟗	1	+7 ↶	✔^(*)	✅	Rate two cross ratios
Ⓢ	1	+1	✘	✅	Steiner Power
⦻	1	+1	✘	✅	Cross Tangent Distance
∠	1	+1	✘	✅	ArcCos of Cycle Angle
⩈	0	+1	✘	✅	Tangent Both
⋐	0	+1	✘	✅	Tangent Inner
⫓	0	+1	✘	✅	Tangent Outer

^(*) Note: the relation may be non-linear if the new cycle is substituted more than once in it.

2.4.2 Constructing relations properties

None of constructing relations may link a new cycle to itself. The required numbers of real parameters, the numbers of other cycles involved and metric independence are as follows:

	Par	Cycles	Metric?	Name
⧉	8	+1	✅	Fractional-linear map
⊞	4	+1	✅	SL(2,R) Moebius map
⊚	4	+2	✅	Ad-hoc cycle reflection
⟃	0	+2	✅	Named cycle conjugation
#	1	+1		Instance with index

2.4.3 Choosing relations properties

Last five choosing relations may link the new cycle to itself (indicated by ↶), thus they appear in New cycle ↷ ToolBar button only. Last four relations do not require any other cycle.

	Par	Cycles	Metric?	Name
≠	0	+1		Different
≄	0	+1		Almost Different
±	1	+1 ↶	✅	Product sign
½	0	+0 ↶		Only Numeric coefficients
ℜ	0	+0 ↶		Only Real coefficients
⇧	0	+0 ↶		Positive orientation
⇩	0	+0 ↶		Negative orientation

2.5 Examples of relations for some common configurations

Some common geometrical configurations can be defined through a single list of cycle relations. We provide below few examples. Here @ refers to the newly created cycle.

Configuration	List of cycle relations
Circle passing three points A, B, C	orthogonal to A orthogonal to B orthogonal to C
Straight line passing two points A, B	orthogonal to A orthogonal to B orthogonal to ∞
Intersection point of lines/circles a, b	orthogonal to a orthogonal to b orthogonal to itself for lines only: different from ∞
Perpendicular from point A to line a	orthogonal to A orthogonal to a orthogonal to ∞
Line from point A parallel to line a	orthogonal to A orthogonal to ∞ tangent to a
Midpoint of interval [A,B]	orthogonal to line AB orthogonal to itself value of cross ratio ⟨ @, ∞; A, B ⟩ =1 different from ∞
Circle with centre O passing point A	in pencil spanned by O and ∞ orthogonal to A
Centre of circle a	in pencil spanned by a and ∞ orthogonal to itself different from ∞
Perpendicular line bisecting segment [A,B]	in pencil spanned by A and B orthogonal to ∞
Bisector of angle between lines a, b	in pencil spanned by a and b rate of cross ratios ⟨ @, a; a, @ ⟩ = ⟨ @, b; b, @ ⟩

An interesting investigation is to see what will be a result of the above constructions if certain points and lines are replaced by various cycles (circles).