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Geometry > Geometry Michealmas > Flashcards

Geometry Michealmas Flashcards

1
Q

A and B are on the same side of l if

A

A = B or segment AB do not intersect l, A,B|*. Otherwise opposite sides of l - a=A|B

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2
Q

Pasch’s Theorem

A

Given a triangle ABC, line l and points A,B,C not on l. If l intersects AB then l intersects either AC or BC.

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3
Q

if l enters triangle through vertex C it intersects

A

AB

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4
Q

Isometry of Euclidean plane is a

A

distance-preserving transformation of E^2. d(f(A),f(B)) = d(A,B)

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5
Q

4 group properties

A

closedness, associativity, identity, inverse

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6
Q

3 isometry theorems

A

every isometry is a 1 to 1 map, composition of any 2 isometries is an isometry, isometries of E^2 form a group with composition as a group operation

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7
Q

4 examples of isometries

A

translation, reflection, rotation, glide reflection

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8
Q

isometry is orientation preserving if the triangle

A

is labelled clockwise. Otherwise, orientation reversing

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9
Q

Isometries orientation-preserving or reversing

A

Translation and rotation are preserving. Reflection and glide reflection are reversing

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10
Q

composition of 2 orientation preserving isometries is

A

orientation preserving

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11
Q

composition of 2 orientation reversing isometries is

A

orientation preserving

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12
Q

composition of an orient preserving and an orient reversing is

A

orient reversing

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13
Q

orientation preserving isometries form a

A

subgroup of Isom(E^2)

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14
Q

Triangles are congruent if

A

lengths and angles are equal

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15
Q

if two congruent triangles then there exists a

A

unique isometry sending A to A’, B to B’ and C to C’

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16
Q

Every isometry is a composition of at

A

most 3 reflections

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17
Q

every non-trivial isometry is one of

A

reflection, rotation, translation, glide reflection

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18
Q

set of fixed points of f Isom(E^2) is

A

Fix(f) = {x in E^2 | f(x) = x}

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19
Q

isometry preserving the origin is a

A

composition of at most 2 reflections

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20
Q

a linear map f(x) = Ax is an isometry iff

A

A is in O(2) an orthogonal subgroup of GL(2,R)

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21
Q

every isometry of E^2 may be written as

A

f(x) = Ax +t

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22
Q

f isometry is orientation preserving if det A =

A

1 and orientation reversing if det A = -1

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23
Q

if A,C is on l then l gives

A

the shortest path from A to C

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24
Q

an action is discrete if none of its orbits possesses

A

accumulation points.

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25
Q

An open connected set F is a fundamental domain for an action G if the sets gF, g in G satisfy

A

the closure of gF in X, for all g in G g is not the identity and the different tiles do not intersect each other and there are finitely many g in G.

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26
Q

if two distinct planes have a common point then they

A

intersect by a line containing that point

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27
Q

if two distinct lines have a common point

A

there exists a unique plane containing both lines

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28
Q

for every triple of non-collinear points there exists

A

a unique plane through these points

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29
Q

given a plane, a point not on the plane and a point on the plane. The two points = distance between plane and point not on plane iff

A

The segment connecting two points is orthogonal to l for all l on the planer and all points on the plane on l This is an orthogonal projection of point not on plane to plane

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30
Q

angle between 2 intersecting planes is the

A

angle between their normals

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31
Q

2 intersecting lines in a plane, A is intersection and line a, A is on a then if

A

a is orthogonal to both intersecting lines then a is orthogonal to the plane

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32
Q

Theorem of 3 perpediculars

A

l is a line on a plane and, B is not on plane, A is on plane and C is on line all points. If BA is orthogonal to plane and AC is orthogonal to line then BC is orthogonal to line

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33
Q

Antipodal

A

diametrically opposite sides

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34
Q

distance between two points on a sphere is

A

radius x angle between points

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35
Q

distance turns into a metric space if these 3 properties hold

A

positivity, symmetry and triangle inequality

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36
Q

curve in a metric space is a geodesic if the curve is

A

locally the shortest path between its points

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37
Q

geodesic, gamma, is closed if

A

thers exists a T where gamma (t) = gamma (t + T) for all t

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38
Q

if all geodesics are open each segment is the

A

shortest path

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39
Q

if all geodesics are closed the shortest path is

A

one of the two segments

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40
Q

every line on a sphere intersects every other line in exactly

A

2 antipodal points

41
Q

angle between 2 line are the angle between the

A

corresponding planes

42
Q

for every line and a point on the line in this line there exists a

A

unique line orthogonal to the line and passing through the point

43
Q

for every line l and a point A not on l where d(A,L) is not pi/2 there exists a unique line l’

A

orthogonal to l and passing through A

44
Q

A triangle on a sphere is a union of …….and a triple of the

A

3 non-collinear point ……shortest paths between them

45
Q

line l = intersection on sphere and corresponding plane through origin. The pole to the line l is the

A

pair of endpoints of the diameter DD’ orthogonal to the plane

46
Q

if a line l contains a point A then the line Pol(A) contains both points of

A

Pol(l)

47
Q

polar correspondence transforms points into ….. and lines into…

A

lines, points

48
Q

a triangle A’B’C’ is polar to ABC if

A

A’ = pol(BC) and angle AOA’<= pi/2 and same for B’ and C’

49
Q

Bipolar theorem if A’B’C’ = Pol(ABC) then

A

ABC = Pol(A’B’C’). If A’B’C’ = Pol(ABC) and triangle ABC has angles alpha, beta, gamma and lengths a, b, c and triangle A’B’C’ has angle pi - a, pi-b, pi-c and lengths pi-alpha, pi-beta, pi - gamma

50
Q

asas, asa, ss HOLD FOR

A

spherical triangles

51
Q

in triangle abc if ab = bc then

A

angle bac = angle bca and m is midpoint of ac then bm is orthogonal to ac

52
Q

AAA holds for

A

spherical triangles

53
Q

thales theorem

A

a = b then angle a = angle b

54
Q

for any spherical triangle

A

angle bisectors are concurrent, perpendicular bisectors, median, altitudes are concurrent and the exist a unique inscribed and a i=unique circumscribed circles for the triangle

55
Q

no domain on S^2 is isometric to a domain on

A

E^2

56
Q

every non-trivial isomtery of S^2 preserving 2 non - antipodal points A,B is a

A

reflection with respect to the line AB

57
Q

given point A,B,C satisfying AB = AC there exists a reflection r such that

A

r(A) = A, r(B) = C , r(C) = B

58
Q

Isometry 6 points:

A

they’re uniquely determined by images of 3 non-collinear points.
isometry act transitively on points of S^2 and on flags in S^2.
The group Isom(S^2) is generated by reflections.
Every isometry of S^2 is a composition of at most 3 reflections.
Every orient preserving isometry is a rotation
every orientation reversing isometry is either reflection or a glide reflection

59
Q

r1, r2, r3 are 3 distinct reflections not preserving the same point on S^2 then

A

r3 o r2 o r1 is a glide reflection

60
Q

Rotations by the same angle are conjugate in

A

Isom(S^2)

60
Q

every 2 reflections are conjugate in

A

Isom(S^2)

61
Q

Similarity group is a group generated by all

A

Euclidean isometries and scalar multiplicaitons. its elements can change size but preserve angles, proportionality of all segments, parallelism and similarity of triangles

62
Q

Affine transformations are all transformations of the form

A

f(x) = Ax + b. they form a group

63
Q

Affine transformations preserve

A

collinearity of points, parallelism of lines, ratios of lengths on any line, concurrency of lines and ratio of areas of triangles

64
Q

Affine transformations act …. on triangles in R^2

A

transitively

65
Q

Affine transformation is uniquely determined by

A

images of 3 non-collinear points

66
Q

medians of triangles in E^2

A

are concurrent

67
Q

every bijection f;R^2 -> R^2 is an affince map if it preserves

A

collinearity of points, betweenness and parallelism

68
Q

if a bijection preserves collinearity then it preswrves

A

parallelism and betweenness

69
Q

the fundamental theorem of affine geometry

A

every bijection preserving collinearity of points is an affine map

70
Q

if f is a bijection which takes circles to circles then f is an

A

affine map

71
Q

every parallel projection is an affine map but not every

A

affine map is a parallel projection

72
Q

points of the projective line are lines through the

A

origin

73
Q

projections preserve

A

cross-ratio of points

74
Q

cross-ration of four lines lying in one plane and passing through one point is the

A

cross-ratio of the four points at which these lines intersect an arbitrary line l

75
Q

any compositions of projections is a

A

linear-fractional map

76
Q

a composition of projections preserving 3 points is an

A

identity map

77
Q

given a, b, c on l and a’, b’, c’ on l’ thers exists a

A

composition of projections which takes a, b, c to a’, b’, c’

78
Q

projective transformations are composition of projections mean

A

projective transofmraitons are linear- fractional transformations

79
Q

a projective transformation of a line is determined by

A

images of 3 points

80
Q

projective transformations preserve

A

cross-ratio of 4 collinear points

81
Q

a triangle is a

A

triple of 3 non-collinear points

82
Q

all traingles are equivalent under

A

projective transformations

83
Q

for any quadliteral there exists a unique projective transformation which takes

A

Q to a quadliteral Q’

84
Q

a bijective map preserving projective lines is a

A

projective map

85
Q

a projection of a plane to another plane is a

A

projective map

86
Q

a projection of a plane to another plane is not an affine map if

A

the planes are not parallel

87
Q

desargues theorem

A

suppose the lines joining the corresponding vertices of triangles A1A2A3 and B1B2B3 intersect at one point S. Then intersection points P1 = A2A3 n B2B3, P2 = A1A3 n B1B3 and P3 = A1A2 n B1B2 are collinear

88
Q

geometry of RP^2 with spherical metric is called elliptic geometry and has properties:

A

for any 2 distinct points there exists a unique line through these points,
any distinct line intersect at a unique point,
for any line l and point p there exists a unique line l’ such that p is on l’ and l is orthogonal l’.
group of isometrise acts transitively on the points of this geometry

89
Q

klein model

A

interior of the unit disc line are chords

90
Q

there exists a projective transformation of the plane that

A

maps a given disc to itself, preserves cross-ratios or collinear points and maps the centre of the disc to an arbitary inner point of the disc

91
Q

isometris act transitively on the

A

points and flags of Klein Model

92
Q

l and l’ be 2 intersecting lines in the Klein model. Let t1 and t2 be tangent lines to the disc at the endpoints of l. then

A

l is orthognal to l’ means t1 intersects with t2 on l’

93
Q

two lines in hyperbolic geometry are called intersecting if

A

they have a common point inside hyperbolic plane

94
Q

two lines in hyperbolic geometry are called parallel if

A

they have a common point on the boundary of hyperbolic plane

95
Q

if two lines in hyperbolic geomtery arenot intersecting or parallel they are

A

divergent or ultra-parallel

96
Q

any pair of divergent lines has a

A

unique common perpendicular

97
Q

An isometry of E^2 preserving a line pointwise is either

A

identity or reflection.

Geometry (2 decks)

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