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Fenchel W. — Elementary geometry in hyperbolic space
Fenchel W. — Elementary geometry in hyperbolic space

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Название: Elementary geometry in hyperbolic space

Автор: Fenchel W.


There exist many excellent books on non-Euclidean geometry. To add another one is motivated by the fact that these books contain very little about the geometry in hyperbolic space which has found various applications. Most of what is known is to be found in very old, not easily accessible papers. Many of these have shortcomings. In general the proofs do not cover exceptional cases, often of
There exist many excellent books on non-Euclidean geometry. To add another one is motivated by the fact that these books contain very little about the geometry in hyperbolic space which has found various applications. Most of what is known is to be found in very old, not easily accessible papers. Many of these have shortcomings. In general the proofs do not cover exceptional cases, often of special interest. Sometimes one refers to a passage to the limit, an exact proof of which may be tedious. There are also cases in which it is not even obvious what can be expected to be valid in the limit. Further, it had already been noted in old papers that satisfactory results in the geometry of lines, which actually covers the elementary geometry of the hyperbolic plane, can only be obtained if oriented lines are considered. The proofs applied yield however often only squares of the relations aimed at, and the determination of the correct signs is not convincing.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 237

Добавлена в каталог: 17.11.2013

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Предметный указатель
$\mathbb{J}$-quaternion      3
Adjugate matrix      8
Altitude line of a hexagon      82
Altitude of a hexagon      82
Amplitudes of a hexagon      102
Amplitudes of a tetrahedron      167
Amplitudes of a triangle      105
Anti-inversion      22
Axial plane of an equidistant surface      39 58
Axis of a motion      45
Axis of an equidistant curve      59
Ball      19
Basic flag      17
Bisector axis      111
Bisector, concordant      72 111
Bisector, reverse      72 111
Bundle of lines, elliptic      75
Bundle of lines, hyperbolic      75
Bundle of lines, parabolic      75
Bundle of lines, planar      75
Bundle of planes, elliptic      34 161
Bundle of planes, hyperbolic      34 160
Bundle of planes, parabolic      34 162
Bundle of spherical surfaces, elliptic      194
Bundle of spherical surfaces, hyperbolic      196
Bundle of spherical surfaces, parabolic      194
CAP      19
Centre of a horosphere      46
Centre of a horosphere of a point-reflection      50
Centroid of a triangle      125
Circular flag      19
Circumscribed circle      118
Circumscribed equidistant curve      118
Circumscribed hexagon      136
Circumscribed horocycle      118
Circumscribed triangle      136
Co-altitude line      131
Co-transversal      108
Complex cylinder      3
Concordant bisector      72 111
Conjugate of a quaternion      1
Convex hull      29
Convex set      29
Cross ratio      11
Cylinder, equidistant      59
Cylindric coordinates      205
Diameter of a horosphere      46
Diameter of an equidistant surface      58
Diametral plane of a horosphere      46
Diametral plane of an equidistant surface      58
Disk      19
Displacement of a motion      46
Double cross      67
Elliptic bundle of lines      75
Elliptic bundle of planes      34 161
Elliptic bundle of spherical surfaces      194
Elliptic net of spherical surfaces      197
Elliptic pencil of lines      72
Elliptic pencil of planes      33 158
Elliptic pencil of spherical surfaces      194
End of a line      28
Equidistant curve      59
Equidistant cylinder      59
Equidistant surface      39 58 176
Ex-centroid of a triangle      125
Excribed circle      120
Excribed equidistant curve      120
Excribed horocycle      120
Extended complex cylinder      3
Exterior of a horosphere      57
Exterior of an equidistant surface      58
Face amplitude of a tetrahedron      167
Flag      17
Flag, circular      19
Flag, spherical      19
Glide reflection      52
Half-space      28
Half-turn      46 55
Harmonic pair of points      15
Horizon of a plane      28
Horizon of an equidistant surface      58
Horocycle      46 57
Horosphere      46 56 176
Horospherical coordinates      204
Hyperbolic bundle of lines      75
Hyperbolic net of spherical surfaces      197
Hyperbolic of planes      34 161
Hyperbolic of spherical surfaces      196
Hyperbolic pencil of lines      72
Hyperbolic pencil of planes      33 157
Hyperbolic pencil of spherical surfaces      195
Improper axis of a parallel motion      47
Improper line      31
Improper plane      31
Improper point      28
Initial point of an oriented line      28
Inner product of quaternions      2
Inscribed circle of a triangle      119
Interior of a horosphere      57
Interior of an equidistant surface      58
inversion      21
Isometry group of a plane      54
Isometry group of Hyperbolic space      44
Law of Cosines      83
Law of cotangents      84
Law of Sines      83
Limit rotation      46
Line      28
Line matrix      61
Line reflection      55
Linear family of spherical surfaces      191
Lobatcefskii function      218
Median      124
Median axis      124
Mirror of a glide reflection      52
Mirror of a parallel reflection      52
Mirror of a plane-reflection      50
Mirror of a rotary reflection      53
Moebius group      22
Moment of two lines      169
Motion      45
Multiplier      13
Net of sperical surfaces      197
normal      31
Normalized line matrix      63
Normalized plane matrix      143
Normalized point matrix      141
Oriented line      64
Oriented plane      143
Orthoaxis      128
Orthogonal frame      34
Orthoscheme      213
Parabolic bundle of lines      75
Parabolic bundle of planes      34 162
Parabolic bundle of spherical surfaces      194
Parabolic net of spherical surfaces      197
Parabolic pencil of lines      72
Parabolic pencil of planes      33 157
Parabolic pencil of spherical surfaces      194
Parallel angle      92
Parallel line and plane      29
Parallel lines      29
Parallel motion      46 55
Parallel planes      29
Parallel reflection      52
Parametric representation of a line      38
Pencil of lines, elliptic      72
Pencil of lines, hyperbolic      72
Pencil of lines, parabolic      72
Pencil of planes, elliptic      33 158
Pencil of planes, hyperbolic      33 157
Pencil of planes, parabolic      33 157
Pencil of spherical surfaces, elliptic      194
Pencil of spherical surfaces, hyperbolic      195
Pencil of spherical surfaces, parabolic      194
Pentagon with four right angles      81
Pentagon, right-angled      80
Planar bundle of lines      75
Plane      28
Plane matrix      140
Plane-reflection      50
Point matrix      140
Point-reflection      50
Polar coordinates      204
Power of a point      183
Quadrangle with two adjacent right angles      81
Quadrangle with two opposite right angles      81
Radical plane      185
Radius of a hexagon      112
Rectangular coordinates      205
Reversal      48
Reverse bisector      72 108
Right-angled hexagon      79
Right-angled pentagon      80
Rotary reflection      53
Rotation      46 55
Side amplitude of a triangle      105
Side-line of a hexagon      79
Skew lines      29
Skrew motion      46
Sphere      56 175
Spherical flag      19
Spherical surface      56 175
Spherical triangle      81 93
Terminal point of an oriented line      28
Translation      46 55
Transversal of a double cross      70
Transversal of a hexagon      108
Ultraparallel line and plane      29
Ultraparallel lines      29
Ultraparallel planes      29
Vertex amplitude of a tetrahedron      167
Vertex amplitude of a triangle      105
Width of a double cross      67
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