Geometry Michealmas Flashcards
A and B are on the same side of l if
A = B or segment AB do not intersect l, A,B|*. Otherwise opposite sides of l - a=A|B
Pasch’s Theorem
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.
if l enters triangle through vertex C it intersects
AB
Isometry of Euclidean plane is a
distance-preserving transformation of E^2. d(f(A),f(B)) = d(A,B)
4 group properties
closedness, associativity, identity, inverse
3 isometry theorems
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
4 examples of isometries
translation, reflection, rotation, glide reflection
isometry is orientation preserving if the triangle
is labelled clockwise. Otherwise, orientation reversing
Isometries orientation-preserving or reversing
Translation and rotation are preserving. Reflection and glide reflection are reversing
composition of 2 orientation preserving isometries is
orientation preserving
composition of 2 orientation reversing isometries is
orientation preserving
composition of an orient preserving and an orient reversing is
orient reversing
orientation preserving isometries form a
subgroup of Isom(E^2)
Triangles are congruent if
lengths and angles are equal
if two congruent triangles then there exists a
unique isometry sending A to A’, B to B’ and C to C’
Every isometry is a composition of at
most 3 reflections
every non-trivial isometry is one of
reflection, rotation, translation, glide reflection
set of fixed points of f Isom(E^2) is
Fix(f) = {x in E^2 | f(x) = x}
isometry preserving the origin is a
composition of at most 2 reflections
a linear map f(x) = Ax is an isometry iff
A is in O(2) an orthogonal subgroup of GL(2,R)
every isometry of E^2 may be written as
f(x) = Ax +t
f isometry is orientation preserving if det A =
1 and orientation reversing if det A = -1
if A,C is on l then l gives
the shortest path from A to C
an action is discrete if none of its orbits possesses
accumulation points.
An open connected set F is a fundamental domain for an action G if the sets gF, g in G satisfy
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.
if two distinct planes have a common point then they
intersect by a line containing that point
if two distinct lines have a common point
there exists a unique plane containing both lines
for every triple of non-collinear points there exists
a unique plane through these points
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
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
angle between 2 intersecting planes is the
angle between their normals
2 intersecting lines in a plane, A is intersection and line a, A is on a then if
a is orthogonal to both intersecting lines then a is orthogonal to the plane
Theorem of 3 perpediculars
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
Antipodal
diametrically opposite sides
distance between two points on a sphere is
radius x angle between points
distance turns into a metric space if these 3 properties hold
positivity, symmetry and triangle inequality
curve in a metric space is a geodesic if the curve is
locally the shortest path between its points
geodesic, gamma, is closed if
thers exists a T where gamma (t) = gamma (t + T) for all t
if all geodesics are open each segment is the
shortest path
if all geodesics are closed the shortest path is
one of the two segments
every line on a sphere intersects every other line in exactly
2 antipodal points
angle between 2 line are the angle between the
corresponding planes
for every line and a point on the line in this line there exists a
unique line orthogonal to the line and passing through the point
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’
orthogonal to l and passing through A
A triangle on a sphere is a union of …….and a triple of the
3 non-collinear point ……shortest paths between them
line l = intersection on sphere and corresponding plane through origin. The pole to the line l is the
pair of endpoints of the diameter DD’ orthogonal to the plane
if a line l contains a point A then the line Pol(A) contains both points of
Pol(l)
polar correspondence transforms points into ….. and lines into…
lines, points
a triangle A’B’C’ is polar to ABC if
A’ = pol(BC) and angle AOA’<= pi/2 and same for B’ and C’
Bipolar theorem if A’B’C’ = Pol(ABC) then
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
asas, asa, ss HOLD FOR
spherical triangles
in triangle abc if ab = bc then
angle bac = angle bca and m is midpoint of ac then bm is orthogonal to ac
AAA holds for
spherical triangles
thales theorem
a = b then angle a = angle b
for any spherical triangle
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
no domain on S^2 is isometric to a domain on
E^2
every non-trivial isomtery of S^2 preserving 2 non - antipodal points A,B is a
reflection with respect to the line AB
given point A,B,C satisfying AB = AC there exists a reflection r such that
r(A) = A, r(B) = C , r(C) = B
Isometry 6 points:
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
r1, r2, r3 are 3 distinct reflections not preserving the same point on S^2 then
r3 o r2 o r1 is a glide reflection
Rotations by the same angle are conjugate in
Isom(S^2)
every 2 reflections are conjugate in
Isom(S^2)
Similarity group is a group generated by all
Euclidean isometries and scalar multiplicaitons. its elements can change size but preserve angles, proportionality of all segments, parallelism and similarity of triangles
Affine transformations are all transformations of the form
f(x) = Ax + b. they form a group
Affine transformations preserve
collinearity of points, parallelism of lines, ratios of lengths on any line, concurrency of lines and ratio of areas of triangles
Affine transformations act …. on triangles in R^2
transitively
Affine transformation is uniquely determined by
images of 3 non-collinear points
medians of triangles in E^2
are concurrent
every bijection f;R^2 -> R^2 is an affince map if it preserves
collinearity of points, betweenness and parallelism
if a bijection preserves collinearity then it preswrves
parallelism and betweenness
the fundamental theorem of affine geometry
every bijection preserving collinearity of points is an affine map
if f is a bijection which takes circles to circles then f is an
affine map
every parallel projection is an affine map but not every
affine map is a parallel projection
points of the projective line are lines through the
origin
projections preserve
cross-ratio of points
cross-ration of four lines lying in one plane and passing through one point is the
cross-ratio of the four points at which these lines intersect an arbitrary line l
any compositions of projections is a
linear-fractional map
a composition of projections preserving 3 points is an
identity map
given a, b, c on l and a’, b’, c’ on l’ thers exists a
composition of projections which takes a, b, c to a’, b’, c’
projective transformations are composition of projections mean
projective transofmraitons are linear- fractional transformations
a projective transformation of a line is determined by
images of 3 points
projective transformations preserve
cross-ratio of 4 collinear points
a triangle is a
triple of 3 non-collinear points
all traingles are equivalent under
projective transformations
for any quadliteral there exists a unique projective transformation which takes
Q to a quadliteral Q’
a bijective map preserving projective lines is a
projective map
a projection of a plane to another plane is a
projective map
a projection of a plane to another plane is not an affine map if
the planes are not parallel
desargues theorem
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
geometry of RP^2 with spherical metric is called elliptic geometry and has properties:
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
klein model
interior of the unit disc line are chords
there exists a projective transformation of the plane that
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
isometris act transitively on the
points and flags of Klein Model
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
l is orthognal to l’ means t1 intersects with t2 on l’
two lines in hyperbolic geometry are called intersecting if
they have a common point inside hyperbolic plane
two lines in hyperbolic geometry are called parallel if
they have a common point on the boundary of hyperbolic plane
if two lines in hyperbolic geomtery arenot intersecting or parallel they are
divergent or ultra-parallel
any pair of divergent lines has a
unique common perpendicular
An isometry of E^2 preserving a line pointwise is either
identity or reflection.
