Title: Theory / Unimportant / Math / Geometry/ / Moise{ / Hyperbolic
Tags: récupéré fromImplication typToBinary MissingMedia toBinaryRelation

Name: The Lobachevskian Parallel Postulate
Definition: Given a line \(L\) and a point \(P\) not on \(L\), there are at least two lines which contain \(P\) and are parallel to \(L\).
Page: 139
Chapter: 9

Name: An \(L\)-circle
Definition: A circle orthogonal to \(C\)
Page: 141
Chapter: 9
Variables: Poincaré Model for Lobachevskian Geometry

Name: An \(L\)-line
Definition: the intersection of \(E\) and of an \(L\)-circle
Page: 141
Chapter: 9
Variables: Poincaré Model for Lobachevskian Geometry
\(E\) the interior of \(C\)
Definition2: a diameter of \(E\)
Conjdef: Either

Name: The \(L\)-plane
Definition: \(E\)
Page: 141
Chapter: 9
Variables: Poincaré Model for Lobachevskian Geometry
\(E\) its interior

Definition: The measure of the angles of the rays, at \(A\), tangeant to \(\overrightarrow{AB}_L\) and \(\overrightarrow{AC}_L\).
Page: 141
Chapter: 9
Variables: Poincaré Model for Lobachevskian Geometry
\(A,B,C\in E\) non-collinear
Notation: \(m\angle BAC\)

Name: The Poincaré Model
Definition: A structure \([E,L,d,m]\), with \(L\) the set of \(L\)-lines, and \(d,m\) as defined on it
given: a metric Euclidean space \([S,\mathcal{L,P},d,m]\) and \(C\) a circle fixed in \(S\).
Page: 141
Chapter: 9
Property: A Model of Lobachevskian geometry

Definition: \(\left|\log_\ell\frac{TR/TS}{UR/US}\right|\), with \(R,S\) the intersection of \(C\) with the \(L\)-line containing \(T\) and \(U\).
Page: 141
Chapter: 9
Variables: Poincaré Model for Lobachevskian Geometry
\(T,U\in E\)
Notation: \(d(T,U)\)

Definition: -42 is non-geometrical mathematics
-Ch 11
End of 12
13 apart from .1 and 2(theo1) are already done
14. from 2 theo 2
17 - 23 is Euclidean geometry
Page: -42,142-369
Chapter: 9
Notation: Omitted pages

Question: The set of \(m\angle APD\) such that \(\overrightarrow{PD}\) intersects \(\overrightarrow {AB}\) is
Answer: non-empty and contains an upper bound.
Page: 371
Chapter: 24
Section: 1
Assuming: \(A,B,P\in S\) not collinear

Name: The critical number for \(P\) and \(\overrightarrow{AB}\)
Definition: The supremum of \(\{r\mid \overrightarrow{PD}\text{ intersects }\overrightarrow{AB}\}\) for \(D\) in the interior of \(\angle PAB\).
Page: 371
Chapter: 24
Section: 1
Assuming: \(A,B,P\in S\) not collinear

Name: The angle of parallelism \(P\) and \(\overrightarrow{AB}\)
Definition: The angle \(APD\) with \(m\angle APD=r_0\) the critical number
Page: 371
Chapter: 24
Section: 1
Assuming: \(A,B,P\in S\) not collinear

Smaller0: \(m\angle APD=r_0\)
Greater0: \(\overrightarrow{PD}\) does not intersect \(\overrightarrow{AB}\)
Page: 371
Chapter: 24
Section: 1
Kind: Theo
Assuming: \(A,B,P\in S\) not collinear
Construction: Suppose \(Q=\overrightarrow{PD}\cap\overrightarrow{AB}\), let \(R\) such that \(A-Q-R\), then \(m\angle APR>m\angle APQ=r_0^\circ\)

Name: An interior ray of \(\angle APD\)
Definition: A ray \(\overrightarrow{AF}\) such that \(F\) is in the interior of \(\angle APD\).
Page: 372
Chapter: 24
Section: 1

Smaller0: \(AP=A'P'\)
Greater0: \(r_0=r_0'\)
Page: 372
Chapter: 24
Section: 1
Variables: \(\angle P^iA^iB^i\) rights
Index: 3
Kind: Theo

Name: The critical function
Definition: The function associating to a length \(a\) the critical angle for \(P,A,B\), with \(PA=a\) and \(\angle PAB\) right
Page: 373, 374
Chapter: 24
Section: 1
Notation: \(c\)
Index: 4, 5, 6
Kind: Theo
Typ: non-increasing.
Property: Either 90 constant, or \(\mathcal A\to [0,90)\)

Smaller0: A point at a unique parallel to a line
Greater0: each point as a unique parallel to a line
Page: 375
Chapter: 24
Section: 1
Index: 7
Kind: Theo
Smaller to greater: iff
Name: All-or-None Theorem
Extra: Theorem is only an implication

Name: An open triangle
Definition: no three points colinear, \(A\), \(B\), \(D\) and \(P\)
Page: 376
Chapter: 24
Section: 2
Definition2: \(B\) and \(D\) are on the same side of \(\overleftrightarrow{AP}\)
Definition3: \(\overleftrightarrow{AB}||\overleftrightarrow{PD}\)
DefType: \(\overright{AB}\cup\overright{PD}\cup{\overline{AP}}\)

Definition: \(\cup_i\overrightarrow{B_iA_i}\cup A_1A_2\)
Page: 376
Chapter: 24
Section: 2
Notation: \(\triangle A_1B_1B_2A_2\)
Assuming: -no three points colinear,
-the \(A_i\) are on the same side of \(\overleftrightarrow{B_1B_2}\)
-\(\overleftrightarrow{A_1B_1}||\overleftrightarrow{A_2B_2}\)

Definition: The lines \(\overleftrightarrow{AB}\) does not intersect \(\overleftrightarrow{PD}\)
Page: 376
Chapter: 24
Section: 2
Notation: \(\overleftrightarrow{AB}||\overleftrightarrow{PD}\)

Name: A critically parallel ray to \(\overrightarrow{AB}\)
Definition: A ray \(\overrightarrow{PD}\) such that \(\triangle DPAB\) and each interior ray of \(\angle APD\) intersects \(\overrightarrow{AB}\).
Page: 376-380
Chapter: 24
Section: 2
Variables: \(A,B,P\in S\) not colinear
Notation: A ray \(\overrightarrow r\) such that \(\overrightarrow{r}|\overrightarrow{AB}\)
Index: 3, 4, 5, 7, 8
Kind: , Theo
Typ: An equivalence relation, with congruence \(\sim\)
Cardinal: 1

Smaller0: \(C-P-D\)
Greater0: \(\overrightarrow{CD}|\overrightarrow{AB}\)
Page: 377
Chapter: 24
Section: 2
Variables: \(A,B,P,D\in S\) no three colinear,
\(C\in S\)
Index: 1
Kind: Theo
Smaller1: \(\overrightarrow{PD}|\overrightarrow{AB}\)

Page: 378
Definition: The symmetric closure of the inclusion relation over rays
Chapter: 24
Section: 2
Notation: \(\sim\)
Name: Equivalence
Context: rays
Equivalence: Equivalence relation

Name: Equivalent open triangles
Definition: Open triangles whose sides's ray are equivalent
Page: 378
Chapter: 24
Section: 2
Index: 5
Kind: theo

Name: An isoceles open triangle
Definition: An open triangles whose angles are congruents.
Page: 378
Chapter: 24
Section: 2

Smaller0: \(\overrightarrow{PD}|\overrightarrow{AD}\)
Greater0: \(\triangle DPAB\) is equivalent to an isoceles open triangle which has \(P\) as a vertex
Page: 378
Chapter: 24
Section: 2
Variables: \(A,B,P,D\)
Index: 6
Kind: Theo

Name: A closed triangle
Definition: An open triangle whose parallel sides are critically parallel
Page: 382
Chapter: 24
Section: 3

Smaller0: \(\triangle BAPD\) is a closed triangle
Greater0: \(\triangle BAPD\) is an open triangle
Page: 382
Chapter: 24
Section: 3

Name: Hyperbolic parallel postulate
Definition: Given a line \(L\) and a point \(P\) there is at least two parallel to \(L\) through \(P\)
Page: 382
Chapter: 24
Section: 3
Abbreviation: HPP

Smaller0: \(m I\)
Greater0: \(m E\)
Page: 382
Chapter: 24
Section: 3
Variables: \(\triangle BAPD\) a closed triangle,
\(E\) an exterior angle,
\(I\) its remote interior angle
Smaller to greater: \(\le\)
Assuming: HPP
Name: The Exterior Angle Theorem
Greater to smaller: \(\ge\)

Smaller0: HPP
Greater0: strictly decreasing
Page: 384
Chapter: 24
Section: 3
Index: 2
Kind: Theo
Prefix Greater: the critical function \(c\) is

Smaller0: HPP
Greater0: acute
Page: 384
Chapter: 24
Section: 3
Kind: Theo
Prefix Greater: the upper base angles of a Saccheri quadrilateral are

Name: The defect of \(\triangle ABC\)
Definition: \(180-\sum m\angle A-m\angle B-m\angle C\)
Page: 386
Chapter: 24
Section: 4
Notation: \(\delta\triangle ABC\)

Name: Proportional sequences of number
Definition: Sequences \((a_i^j)_i\), for \(j\in\{1,2\}\), such that \(\frac{a^1_i}{a_i^2}\) is constant
Page: 172
Chapter: 12
Section: 1
Context: Vectors
Notation: \(\sim\)
Typ: Symmetric relation

Name: Their constant ratio
Definition: \(\frac{a^1_i}{a_i^2}\) for any \(i\)
Page: 172
Chapter: 12
Section: 1
Variables: \((a_i^j)_i\) proportional sequences

Definition: The length of the side opposite to \(A_i\)
Page: 174
Chapter: 12
Section: 2
Variables: \(\triangle A_1A_2A_3\)
Notation: \(a_i\)

Name: Similarity between triangles
Definition: Two triangles whose corresponding sides are proportional and corresponding angles are congruent
Page: 174
Chapter: 12
Section: 2
Typ: Equivalence relation

Definition: The measure of a base angle of a right isoceles triangle with leg of size \(a\)
Page: 389
Chapter: 24
Section: 4
Variables: \(a>0\)
Notation: \(r(a)\)

Question: The set of possible defect is
Answer: \((0,180)\)
Page: 389, 405
Chapter: 24
Section: 4, 9
Index: 4, 1
Kind: Theo, Theo

Name: A subdivision of \(K\)
Definition: A union of a triangulation of each elements of \(K\).
Page: 390
Chapter: 24
Section: 5
Variables: \(K\) a complex
Typ: Transitive relation

Name: The interior of \(R\)
Definition: The intersection of the \(\angle P_{i-1}P_iP_{i+1}\), where \((P_i)\)
Page: 390
Chapter: 24
Section: 5
Variables: \(R=\)
Assuming: \(R\) is convex

Name: A convex polygonal region
Definition: The union of a convex polygon and of its interior.
Page: 390
Chapter: 24
Section: 5

Name: A star triangulation
Definition: A triangulation whose every triangle have a vertex in common.
Page: 390
Chapter: 24
Section: 5
Variables: \(\mathcal R\) a convex polygonal region

Question: \(K_1\) and \(K_2\) have
Answer: a common subdivision
Page: 391
Chapter: 24
Section: 5
Variables: (R=\)
\(K_1,K_2\) two triangulations of \(R\)
Index: 2
Kind: Theo

Page: 392
Definition: The relation between polygons \(R^i\) which have triangulations \((T^i_j)_{j}\) such that \(T^1_j\cong T^2_j\) (i.e. triangles are congruent).
Chapter: 24
Section: 5
Notation: \(\equiv\) on polygons
Name: Equivalence by finite decomposition
Kind: Theo
Index: 3
Equivalence: yes

Smaller0: The area of \(R_1\) and \(R_2\) are equal
Greater0: \(R_1\equiv R_2\)
Page: 394, 398, 410
Chapter: 24
Section: 6, 10
Variables: \(R_1,R_2\) polygonal regions
Index: 5, 1
Kind: Theo, Theo
Smaller to greater: iff
Name: Bolyai's Theorem
Extra: \(R_1\equiv R_2\)  means they have triangular decomposition \((T_1^i), (T_2^i)\) with \(T_1^i\cong T_2^i\)

Name: The parallelogram associated with \(\triangle ABC\)
Definition: Take \(D\) the midle of \(AB\)
Take \(E\) the middle of \(AC\)
Take \(G\) such that \(E\) is the middle of \(DG\)
Page: 395
Chapter: 24
Section: 6
Intuition: A parallelegram with same base, partial side in common, and same area
Assuming: The foot of the perpendicular to\(\overline{BC}\) is on the side of \(B\)

Name: The rectangle associated with \(\triangle ABC\)
Definition: \(\square BCHF\) where:
Take \(D\) the middle of \(AB\)
take \(E\) the middle of \(AC\)
Take \(F\) the projection of \(B\) on \(DE\)
Take \(H\) the projection of \(C\) on \(DE\)
Page: 394, 403
Chapter: 24, 24
Section: 6, 8
Intuition: Same base \(BC\) and same area
Typ: A rectangle

Name: The defect of \(K\)
Definition: \(\sum_{T\in K}\delta T\)
Page: 399
Chapter: 24
Section: 7
Variables: \(K\) a complex
Context: complex
Notation: \(\delta K\)
Assuming: HPP

Question: The defect of \(K_1\) and of \(K_2\) are
Answer: equal
Page: 399
Chapter: 24
Section: 7
Variables: \(R\) a convex polygonal region,
\(K_1,K_2\) two of its star triangulation
Assuming: HPP

Name: Its defect
Definition: The defect of any/all of the star triangulation of \(R\)
Page: 399
Chapter: 24
Section: 7
Variables: \(R\) a convex polygonal region,
Context: convex polygonal region,
Assuming: HPP

Name: A border triangulation
Definition: A star triangulation with the central vertex lying on an edge of the polygon
Page: 399
Chapter: 24
Section: 7
Variables: \(R\) a convex polygonal region,
Index: 24.57
Kind: Fig

Question: \(\sum \delta R_i\)
Answer: equals \(\delta R\)
Page: 400
Chapter: 24
Section: 7
Variables: \(R\) a convex polygonal regions,
\(L\) a line intersecting the interior of \(R\)
\(R_1,R_2\) the regions
Index: 3
Kind: Theo
Assuming: HPP

Question: The defects of the triangulations
Answer: are equal
Page: 400
Chapter: 24
Section: 7
Variables: \(R\) a convex polygonal regions,
\(K_i\) triangulations
Index: 3
Kind: Theo
Assuming: HPP

Name: The defect of \(R\)
Definition: The defect of any/all of triangulations of \(R\)
Page: 401
Chapter: 24
Section: 7
Variables: \(R\) a polygonal region,
Assuming: HPP

Name: The \((P_i^j)_i\) are congruent
Definition: \(\overline{P_i^1P_{i'}^1}\cong\overline{P_i^2P_{i'}^2}\) and
\(\angle{P_i^1P_{i'}^1P_{i''}^1}\cong\angle{P_i^2P_{i'}^2P_{i''}^2}\)  for each \(i,i',i''\)
Page: 402
Chapter: 24
Section: 8
Variables: \((P_i^j)_i\) be polygon
Notation: \((P_i^1)_i\cong(P_i^2)_i\)
Typ: Equivalence relation

Question: The set of functions \(\alpha:\mathcal R\to\mathbb R^{>0}\) such that:
-\(\alpha(R_1\cup R_2)=\alpha R_1+\alpha R_2\) when their intersection are edges and vertices and
-\(R_1\equiv R_2\) implies \(\alpha R_1=\alpha R_2\)
Answer: is the set of multiple of \(\delta\) by a positive real
Page: 413
Chapter: 24
Section: 9
Index: 1
Kind: Theo

Name: A punctured plane
Definition: A plane, minus a point
Page: 417
Chapter: 25
Section: 2

Name: The inversion of \(E-A\) about \(C\)
Definition: [$]\defFun{E-A}{E-A}{P}{P'}[/$] with \(A-P-P'\) and \((AP)(AP')=a^2\)
Page: 417
Chapter: 25
Section: 2
Variables: \(E\) a plane with
\(C=\)
Notation: \(f\)
Index: , 1, 3
Kind: , theo, Theo
Example: \(P\)
Example2: a point close to infinity
Extra: \(E-A\) is the punctured plane \(E\) without \(A\)
Intuition: Polynomial sending circle to itself, center to infinity, smoothly
Applied to: A point \(P\) on \(C\)
Example4: A point in the exterior of \(C\)
Applied to3: \(f(P)\) for some \(P\in E-A\)
Applied to2: A point close to \(A\)
Applied to4: A point in the interior of \(C\)

Question: \(f(L-A)\)
Answer: equals \(L-A\)
Page: 418
Chapter: 25
Section: 2
Variables: \(C=\)
\(L\) a line
Context: inversion of \(E-A\) about \(C\)
Index: 4
Kind: Theo
Assuming: \(A\) lies on \(L\)

Question: \(f(re^{i\theta})\)
Answer: equals \(\frac{a^2}{r}e^{i\theta}\)
Page: 419
Chapter: 25
Section: 2
Variables: \(C=\) with center 0
\(r>0\), \(\theta\in\mathbb R\)
Context: inversion of \(E-A\) about \(C\)

Name: A \(k\)-set
Definition: \(F\setminus\{A\}\) for \(F\) a line or a circle
Page: 419
Chapter: 25
Section: 2
Variables: \(C=\)
Extra: \(F\) may or may not contain \(A\)

Smaller0: For some line and some point, parallels are unique
Greater0: The plane contains at least one rectangle
Page: 415
Chapter: 24
Section: 14
Smaller to greater: iff
Equivalent: there is an area function whose value on triangles depends only on the base and altitude of the triangle
Equivalent3: The plane contains at least one triangle for which the angle is 180°
Equivalent2: The plane contains at least two similar triangles which are not congruent