Abstract
We study some characterizations of hyperbolic geometry in the Poincaré disk. We first obtain the hyperbolic area and length formula of Euclidean disk and a circle represented by its Euclidean center and radius. Replacing interior angles with vertices coordinates, we also obtain a new hyperbolic area formula of a hyperbolic triangle. As its application, we give the hyperbolic area of a Lambert quadrilateral and some geometric characterizations of Lambert quadrilaterals and Saccheri quadrilaterals.
1. Introduction
Hyperbolic geometry was created in the first half of the nineteenth century in order to prove the dependence of Euclid’s fifth postulate on the first four ones. It was extensively studied by Nikolai Lobachevskii and Johann Bolyai. Because of this, hyperbolic geometry is also known as Bolyai-Lobachevskian geometry. The basis necessary for an analytic study of hyperbolic geometry was laid by Leonhard Euler, Gaspard Monge, and Friedrich Gauss in their investigation of curved surfaces. Later, two of the most famous analytic models of the hyperbolic geometry were built, which are known as the Klein model and the Poincaré model in the name of their inventors [1]. One can refer to [2–4] for more history about hyperbolic geometry.
For a complex number , we assume that , where . For convenience, we denote by the coordinate of the point . We denote the complex plane, the unit disk, and the upper half plane by , , and , respectively. We denote by the Euclidean circle with center and radius . Let be the hyperbolic geodesic segment joining and when or , or the hyperbolic geodesic ray with one of the two points and is on or . Let be a simply connected domain of hyperbolic type and let be the hyperbolic metric of with the Gaussian curvature . In particular, the hyperbolic metric density in and with the Gaussian curvature is given by respectively.
Define the hyperbolic area of a measurable subset of as Particularly, if is a measurable subset in , then The integrand at (3) is replaced with when . We also denote the Euclidean area of a measurable subset by . For any rectifiable curve in , the hyperbolic length of is If is or , then the integrands in (4) are and , respectively. The hyperbolic distance between two points is defined by where the infimum is taken over all rectifiable curves in joining and . We say that a curve is a hyperbolic geodesic joining and if, for all , it follows
If there exists a polygon enclosed by hyperbolic geodesics, then we call it a hyperbolic polygon. Particularly, if takes or , then we call it a hyperbolic triangle and hyperbolic quadrilateral, respectively. The interior angle of a hyperbolic polygon denotes the intersectional angle of the tangents of two geodesic arcs at the vertex. If there exists a hyperbolic quadrilateral with angles , , then it is said to be a Lambert quadrilateral [5, p. 156]. If there exists a hyperbolic quadrilateral with angles , , then it is called a Saccheri quadrilateral [5, p. 156].
Given two nonempty subsets of , let denote the hyperbolic distance between them, defined as where stands for the hyperbolic distance between two points and .
We also need the following three explicit formulas: for all . In particular, for ,
The above basic facts can be found in our standard references [5, 6] and in many other sources on hyperbolic geometry such as [7, 8]. Far-reaching and specialized advanced texts discussing hyperbolic geometry include [9, 10].
Hyperbolic triangles and Lambert quadrilaterals are fundamental geometric quantities for the study of hyperbolic geometry theory and its applications. Curien and Werner [11] constructed random triangulations of the Poincaré disc by hyperbolic triangles. Demirel [12], Yang and Fang [13, 14] gave characterizations of Möbius transformations by use of hyperbolic triangles or Lambert quadrilaterals. Pambuccian [15] showed that mappings preserving the area equality of hyperbolic triangles are motions. One can see [16–18] for more characterizations about quasiconformal mappings and harmonic quasiconformal mappings in the sense of hyperbolic metrics.
Further study of geometric properties of hyperbolic triangles, Lambert quadrilaterals, and hyperbolic polygons also raises one's interest. Rostamzadeh and Taherian [19] considered the Klein model of the real hyperbolic plane and gave a new definition of its defect and area. A hyperbolic area formula and the radius of the inscribed circle of a hyperbolic triangle in the Poincaré model can be found in [5, p. 150, 152]. Recently, Vuorinen and Wang [20] obtained sharp bounds for the product and the sum of two hyperbolic distances between two opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Kanesaka and Nakamura [21] gave an explicit hyperbolic area formula for a -acute triangle. One can refer to [22–24] for distortion estimates of hyperbolic areas of measurable subsets under quasiconformal mappings.
By the hyperbolic radius of a disc, the following is shown.
Theorem A (see [5]). (1) The area of a hyperbolic disc of radius is .
(2) The length of a hyperbolic circle of radius is .
Instead of a hyperbolic radius, we use the Euclidean center and radius of Euclidean disc in to give its hyperbolic area formula as follows.
Theorem 1. For Euclidean disc , the hyperbolic area of is where .
Area formula for a hyperbolic polygon can be determined by its interior angles [5, p. 150].
Theorem B. For any hyperbolic triangle with interior angles ,
In this paper, instead of interior angles, we will use the coordinates of vertexes of a hyperbolic triangle to give another hyperbolic area formula. We first study a hyperbolic triangle with a vertex at infinity.
Theorem 2. Let with . (1)If , (2)If , where .
For a general hyperbolic triangle in , its hyperbolic area can be represented by four hyperbolic triangles with a vertex at infinity (see Remark 6).
As some applications of the hyperbolic area formula of a hyperbolic triangle, we first give an explicit formula for the hyperbolic area of a Lambert quadrilateral in Section 3. Moreover, we obtain the length of two hyperbolic diagonal lines in a Lambert quadrilateral in as follows.
Theorem 3. Let be the Lambert quadrilateral in with , . Then where .
Some geometric characterizations of Saccheri quadrilaterals are studied and a relation about two hyperbolic distances between two pairs of opposite sides is given in Section 4 (see Theorem 14).
2. Hyperbolic Areas of Hyperbolic Triangles in H2
Proof of Theorem 1. Let be Euclidean disc in . By direct calculation, the hyperbolic area of is given by Let , which maps onto the disk with center 0 and radius , where Then we have
Lemma 4. Let be a hyperbolic geodesic in with two endpoints , . Then the hyperbolic area of the hyperbolic triangle is given as follows:
Proof.
Case 1. If , then
Particularly, when , then .
Case 2. If , we have
In particular, when , .
Lemma 5. Let be a hyperbolic geodesic with two endpoints , in ; then the hyperbolic area of the hyperbolic triangle is as follows:
Proof. Let . Without loss of generality, we assume .(1)If , by Lemma 4, we have (2)If , using Lemma 4 again, we get (3)If , Lemma 4 also implies Similarly, if we have the same conclusion: Thus, the proof of Lemma 5 is completed.
Theorem 2 is an application of Lemma 5. For the proof of Theorem 2 we also need a sharp formula of the circle orthogonal to the boundary of [25].
Lemma A. Let with . Then is orthogonal to , where
Proof of Theorem 2. We may assume that . Let be the circle which is through the two points , where and are represented as in (27). Then we have , .
Case 1 (). If , then by Lemma 5
If , so we have
Case 2 (). This case implies that ; then
Remark 6. For a general hyperbolic triangle in , we can give its hyperbolic area formula by Theorem 2. Given three points , , and in with , we may assume that . Let be the circle which is through and and orthogonal to . Let be the point which is the intersection of and the hyperbolic geodesic through point and orthogonal to . Then we have
where and .
Thus it follows
where
then by Theorem 2 we can get the result of .
Example 7. Let , , ; by calculation we have where .
Remark 8. There are lots of triangles which have the same hyperbolic area but different Euclidean areas in . For instance, and are two hyperbolic triangles in with , , , , , and
Remark 9. For any hyperbolic -polygons () in , we always can divide it into several hyperbolic triangles ; then we have
Remark 10. For any hyperbolic -polygons () in , since the hyperbolic area is invariant under Möbius transformation, so we can map the into by a Möbius transformation; therefore we obtain the hyperbolic area of in .
3. Hyperbolic Areas of Lambert Quadrilaterals
As an application of the explicit hyperbolic area formula of a hyperbolic triangle, we obtain the hyperbolic area of a Lambert quadrilateral.
Theorem 11. Let be the Lambert quadrilateral with angles , in and , ; then (1)if , (2)if , where , , , and are given at (39).
Proof. Without loss of generality, we assume that , and , (see Figure 1), which implies , . By (27) we have
When , we get
where . So, the following three relations
imply that
When , it follows
and then we get
The proof of the case that is similar to the case that . So the proof of Theorem 11 is complete.
In particular, when , we have
Remark 12. Since the Saccheri quadrilateral can be divided into two Lambert quadrilaterals, therefore the hyperbolic area of a Saccheri quadrilateral is twice of a Lambert quadrilateral.
4. Hyperbolic Geometric Characterization of Lambert Quadrilaterals and Saccheri Quadrilaterals
Lemma B (see [25]). Let be a constant with . Then is orthogonal to for . Given such that 0, , and are noncollinear, the orthogonal circle contains and if and .
The following result in Lemma 13 appeared in [20]; for completeness, we give its proof as follows.
Lemma 13. Let be a Lambert quadrilateral (see Figure 2) in with , and let then where , , .
Proof. Assume that is on the real axis, is on the imaginary axis, and , and . We use (10) to obtain Utilizing the relation (46), we have where denote the center and radius of the circle through the geodesic and and denote the center and radius of the circle through the geodesic . Write and . Let . Then the relation (9) implies that From the relation (7) we know that . The function reaches the minimum value Then where . Similarly, we have where .
Proof of Theorem 3. Assume that and , , and let be on the real axis and on the imaginary axis (see Figure 2). We use (10) to obtain which implies By the relation (46) we have that the geodesic is on the circle , where Similarly, the geodesic is on the circle , where Hence, and . Utilizing the relation (9), we obtain This completes the proof of Theorem 3.
Theorem 14. Let be a Saccheri quadrilateral in with , , and then where .
Proof. Assume that , , and and let be on the real axis and on the imaginary axis (see Figure 3). Then by the relation (46) we have that the circle through points and which is orthogonal to is , where It follows from the relations (10) and (48) that where . Thus, This completes the proof of Theorem 14.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This paper is supported by the Natural Science Foundation of Fujian Province of China (2014J01013), NCETFJ Fund (2012FJ-NCET-ZR05), and the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX110).