![Page 1: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/1.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
What is Hyperbolic Geometry?
Mahan Mj,Department of Mathematics,RKM Vivekananda University.
Mahan Mj
![Page 2: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/2.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 3: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/3.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 4: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/4.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 5: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/5.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 6: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/6.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 7: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/7.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
History
Euclid’s Axioms
1 Any two points in a plane may be joined by a straight line.
2 A finite straight line may be extended continuously in a straightline.
3 A circle may be constructed with any centre and radius.
4 All right angles are equal to one another.
5 If a straight line falling on two straight lines makes the interiorangles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on whichthe angles are less than the two right angles.
Mahan Mj
![Page 8: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/8.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Circa 100 BC:5th postulate is equivalent toThrough a point not on a straight line there is one and only onestraight line through the point parallel to the given straight line.The attempt to prove the 5th postulate from the otherpostulates gave rise to hyperbolic geometry.
Mahan Mj
![Page 9: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/9.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Circa 100 BC:5th postulate is equivalent toThrough a point not on a straight line there is one and only onestraight line through the point parallel to the given straight line.The attempt to prove the 5th postulate from the otherpostulates gave rise to hyperbolic geometry.
Mahan Mj
![Page 10: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/10.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Circa 100 BC:5th postulate is equivalent toThrough a point not on a straight line there is one and only onestraight line through the point parallel to the given straight line.The attempt to prove the 5th postulate from the otherpostulates gave rise to hyperbolic geometry.
Mahan Mj
![Page 11: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/11.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Circa 100 BC:5th postulate is equivalent toThrough a point not on a straight line there is one and only onestraight line through the point parallel to the given straight line.The attempt to prove the 5th postulate from the otherpostulates gave rise to hyperbolic geometry.
Mahan Mj
![Page 12: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/12.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 13: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/13.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 14: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/14.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 15: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/15.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 16: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/16.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 17: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/17.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
HistoryGauss started thinking of parallels about 1792. In an 18thNovember, 1824 letter to F. A. Taurinus, he wrote:‘The assumption that the sum of the three angles (of a triangle)is smaller than two right angles leads to a geometry which isquite different from our (Euclidean) geometry, but which is initself completely consistent.But Gauss did not publish his work.
Mahan Mj
![Page 18: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/18.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
In the 18th century, Johann Heinrich Lambert introduced whatare today called hyperbolic functions and computed the area ofa hyperbolic triangle.In the nineteenth century, hyperbolic geometry was extensivelyexplored by the Hungarian mathematician Janos Bolyai and theRussian mathematician Nikolai Ivanovich Lobachevsky, afterwhom it is sometimes named. Lobachevsky published a paperentitled On the principles of geometry in 1829-30, while Bolyaidiscovered hyperbolic geometry and published his independentaccount of non-Euclidean geometry in the paper The absolutescience of space in 1832.The term "hyperbolic geometry" was introduced by Felix Kleinin 1871.
Mahan Mj
![Page 19: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/19.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
In the 18th century, Johann Heinrich Lambert introduced whatare today called hyperbolic functions and computed the area ofa hyperbolic triangle.In the nineteenth century, hyperbolic geometry was extensivelyexplored by the Hungarian mathematician Janos Bolyai and theRussian mathematician Nikolai Ivanovich Lobachevsky, afterwhom it is sometimes named. Lobachevsky published a paperentitled On the principles of geometry in 1829-30, while Bolyaidiscovered hyperbolic geometry and published his independentaccount of non-Euclidean geometry in the paper The absolutescience of space in 1832.The term "hyperbolic geometry" was introduced by Felix Kleinin 1871.
Mahan Mj
![Page 20: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/20.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
In the 18th century, Johann Heinrich Lambert introduced whatare today called hyperbolic functions and computed the area ofa hyperbolic triangle.In the nineteenth century, hyperbolic geometry was extensivelyexplored by the Hungarian mathematician Janos Bolyai and theRussian mathematician Nikolai Ivanovich Lobachevsky, afterwhom it is sometimes named. Lobachevsky published a paperentitled On the principles of geometry in 1829-30, while Bolyaidiscovered hyperbolic geometry and published his independentaccount of non-Euclidean geometry in the paper The absolutescience of space in 1832.The term "hyperbolic geometry" was introduced by Felix Kleinin 1871.
Mahan Mj
![Page 21: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/21.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
In the 18th century, Johann Heinrich Lambert introduced whatare today called hyperbolic functions and computed the area ofa hyperbolic triangle.In the nineteenth century, hyperbolic geometry was extensivelyexplored by the Hungarian mathematician Janos Bolyai and theRussian mathematician Nikolai Ivanovich Lobachevsky, afterwhom it is sometimes named. Lobachevsky published a paperentitled On the principles of geometry in 1829-30, while Bolyaidiscovered hyperbolic geometry and published his independentaccount of non-Euclidean geometry in the paper The absolutescience of space in 1832.The term "hyperbolic geometry" was introduced by Felix Kleinin 1871.
Mahan Mj
![Page 22: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/22.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 23: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/23.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 24: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/24.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 25: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/25.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 26: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/26.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 27: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/27.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 28: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/28.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 29: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/29.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
The Problem
Parallel PostulateGiven a straight line L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straight line L′
lying on P passing through x and parallel to L.
ProblemProve the Parallel Postulate from the other axioms of Euclideangeometry.
Mahan Mj
![Page 30: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/30.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 31: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/31.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 32: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/32.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 33: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/33.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 34: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/34.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 35: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/35.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
QuestionWhat is:1)a plane P2) a straight line L3) the notion of parallellism?
Mahan Mj
![Page 36: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/36.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 37: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/37.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 38: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/38.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 39: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/39.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 40: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/40.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 41: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/41.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 42: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/42.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 43: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/43.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 44: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/44.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 45: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/45.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Euclidean Geometry Revisited
Answers for Euclidean Geometry:The Euclidean plane is R2 equipped with the metric
ds2 = dx2 + dy2.
(Infinitesimal Pythagoras)Meaning: Lengths of curves σ (= smooth maps of [0,1] intoR2) are computed as per the formulal(σ) =
∫ 10 ds =
∫ 10 [(
dxdt )
2 + (dydt )
2]12 dt (A)
for some parametrization x = x(t), y = y(t) of the curve σ.
Mahan Mj
![Page 46: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/46.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem
Given two points (x1, y1), (x2, y2) ∈ R2, the straight linesegment between (x1, y1) and (x2, y2) is the unique path thatrealizes the shortest distance (as per formula A) between them.
DefinitionTwo bi-infinite straight lines are said to be parallel if they do notintersect.
Mahan Mj
![Page 47: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/47.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem
Given two points (x1, y1), (x2, y2) ∈ R2, the straight linesegment between (x1, y1) and (x2, y2) is the unique path thatrealizes the shortest distance (as per formula A) between them.
DefinitionTwo bi-infinite straight lines are said to be parallel if they do notintersect.
Mahan Mj
![Page 48: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/48.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem
Given two points (x1, y1), (x2, y2) ∈ R2, the straight linesegment between (x1, y1) and (x2, y2) is the unique path thatrealizes the shortest distance (as per formula A) between them.
DefinitionTwo bi-infinite straight lines are said to be parallel if they do notintersect.
Mahan Mj
![Page 49: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/49.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 50: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/50.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 51: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/51.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 52: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/52.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 53: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/53.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 54: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/54.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
General Geometries
Answers for a slightly more general geometry: Metric onU ⊂ R2:
ds2 = f (x , y)dx2 + g(x , y)dy2.
Here, l(σ) =∫ 1
0 ds =∫ 10 [f (x(t), y(t))(
dxdt )
2 + g(x(t), y(t))(dydt )
2]12 dt ....(B)
Mahan Mj
![Page 55: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/55.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 56: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/56.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 57: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/57.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 58: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/58.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 59: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/59.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 60: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/60.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 61: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/61.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Definitions:1) Given two points (x1, y1), (x2, y2) ∈ R2, the geodesicbetween (x1, y1) and (x2, y2) is the unique path that realizes theshortest distance (as per formula B) between them.2) An isometry I is a map that preserves the metric, i.e. ifI((x , y)) = (x1, y1) then
f (x , y)dx2 + g(x , y)dy2 = f (x1, y1)dx21 + g(x1, y1)dy2
1 .
Mahan Mj
![Page 62: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/62.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 63: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/63.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 64: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/64.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 65: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/65.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 66: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/66.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 67: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/67.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 68: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/68.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Hyperbolic Geometry
Answers for hyperbolic geometry:A Model for hyperbolic geometry is the upper half planeH = (x , y) ∈ R2, y > 0 equipped with the metricds2 = 1
y2 (dx2 + dy2).
Where does this come from? Another story.
Mahan Mj
![Page 69: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/69.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 70: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/70.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 71: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/71.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 72: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/72.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 73: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/73.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 74: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/74.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 75: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/75.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 76: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/76.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 77: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/77.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 78: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/78.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 79: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/79.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 80: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/80.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 81: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/81.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 82: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/82.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 83: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/83.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 1:Vertical straight lines in H are geodesics. In fact, the verticalsegment between a,b is the unique geodesic between a,b.Theorem 2:1) Translations: Define f : H→ H by f (x , y) = (x + a, y) forsome fixed a ∈ R.2) Inversions about semicircles: Define g : H→ H by g(z) = R2
zfor some R > 0, where z denotes the complex conjugate of z.Then f ,g are isometries of H.Observation 3:Image of a geodesic under an isometry is another geodesic.Hence images of vertical geodesics under inversions aregeodesics.
Mahan Mj
![Page 84: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/84.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Inversions about semicircles
l(ab) · l(ac) = l(ad)2.
Mahan Mj
![Page 85: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/85.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Inversions about semicircles
l(ab) · l(ac) = l(ad)2.
Mahan Mj
![Page 86: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/86.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 87: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/87.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 88: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/88.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 89: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/89.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 90: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/90.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 91: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/91.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
Theorem 4: In the hyperbolic plane H,vertical straight linesand semi-circular arcs with center on the real axisare geodesics.
Mahan Mj
![Page 92: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/92.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 93: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/93.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 94: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/94.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 95: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/95.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 96: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/96.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 97: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/97.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj
![Page 98: What is Hyperbolic Geometry? - amitava/MMC/1.pdf · Russian mathematician Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On](https://reader033.vdocuments.net/reader033/viewer/2022060913/60a7a3b5fc35be529247e568/html5/thumbnails/98.jpg)
HistoryThe Problem
Euclidean Geometry RevisitedGeneral Geometries
Hyperbolic Geometry
We have the following New Parallel Postulate:Given a geodesic L in H and a point x on H lying outside L,there exist INFINITELY many bi-infinite geodesics L′ lying on Hpassing through x and parallel to L.
Mahan Mj