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Mathematics LibreTexts

17.1: Special bijection on the h-plane

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Consider the conformal disc model with the absolute at the unit circle Ω centered at O. Choose a coordinate system (x,y) on the plane with the origin at O, so the circle Ω is described by the equation x2+y2=1.

截屏2021-03-01 上午10.40.49.png
The plane thru P,O, and S.

Let us think that our plane is the coordinate xy-plane in the Euclidean space; denote it by Π. Let Σ be the unit sphere centered at O; it is described by the equation

x2+y2+z2=1.

Set S=(0,0,1) and N=(0,0,1); these are the south and north poles of Σ.

Consider stereographic projection ΠΣ from S; given point PΠ denote its image in Σ. Note that the h-plane is mapped to the north hemisphere; that is, to the set of points (x,y,z) in Σ described by the inequality z>0.

For a point PΣ consider its foot point ˆP on Π; this is the closest point to P.

Note that the composition PPˆP of these two maps gives a bijection from the h-plane to itself. Further note that P=ˆP if and only if PΩ or P=O.

Exercise 17.1.1

Suppose that PˆP is the bijection described above. Assume that P is a point of h-plane distinct from the center of absolute and Q is its inverse in the absolute. Show that the midpoint of [PQ] is the inversion of ˆP in the absolute.

Hint

截屏2021-03-01 上午11.23.55.png

Let N,O,S,P,P and ˆP be as on the diagram above.

Note that QQ=1x and therefore we need to show that OˆP=2/(x+1x). To do this, show and use that SOPSPNPˆPP and 2SO=NS.

Lemma 17.1.1

Let (PQ)h be an h-line with the ideal points A and B. Then ˆP,ˆQ[AB].

Moreover,

AˆQBˆP^QBˆPA=(AQBPQBPA)2.

In particular, if A,P,Q,B appear in the same order, then

PQh=12lnAˆQBˆPˆQBˆPA.

Proof

Consider the stereographic projection ΠΣ from the south pole S. Note that it fixes A and B; denote by P and Q the images of P and Q;

According to Theorem 16.3.1c,

AQBPQBPA=AQBPQBPA.

By Theorem Theorem 16.3.1e, each circline in Π that is perpendicular to Ω is mapped to a circle in Σ that is still perpendicular to Ω. It follows that the stereographic projection sends (PQ)h to the intersection of the north hemisphere of Σ with a plane perpendicular to Π.

Suppose that Λ denotes the plane; it contains the points A, B, P, ˆP and the circle Γ=ΣΛ. (It also contains Q and ˆQ but we will not use these points for a while.)

截屏2021-03-01 上午11.16.24.png
The plane Λ.

Note that

  • A,B,PΓ,
  • [AB] is a diameter of Γ,
  • (AB)=ΠΛ,
  • ˆP[AB]
  • (PˆP(AB).

Since [AB] is the diameter of Γ, by Corollary 9.8, the angle APB is right. Hence AˆPPAPBPˆPB. In particular

APBP=AˆPPˆP=PˆPBˆP.

Therefore

AˆPBˆP=(APBP)2.

The same way we get that

AˆQBˆQ=(AQBQ)2.

Finally, note that 17.1.2+17.1.3+17.1.4 imply 17.1.1

The last statement follows from 17.1.1 and the definition of h-distance. Indeed,

PQh:=lnAQBPQBPA==ln(AˆQBˆPˆQBˆPA)12==12lnAˆQBˆPˆQBˆPA.

Exercise 17.1.2

Let Γ1, Γ2, and Γ3 be three circles perpendicular to the circle Ω. Let [A1B1], [A2B2], and [A3B3] denote the common chords of Ω and Γ1, Γ2, Γ3 respectively. Show that the chords [A1B1], [A2B2], and [A3B3] intersect at one point inside Ω if and only if Γ1, Γ2, and Γ3 intersect at two points.

截屏2021-03-01 上午11.21.52.png

Hint

Consider the bijection PˆP of the h-plane with absolute Ω. Note that ˆP[AiBi] if and only if PΓi.


This page titled 17.1: Special bijection on the h-plane is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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