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Luca Kollmer
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Decks in this class (30)
1. Introduction
Definitionmetricmetric space,
Lemma 1bilinearity of the inner p...
2
cards
2. Euclidean Metric
Lemma 2cauchy schwartz inequality,
Lemma 3translation invariance of ...,
Lemma 4triangle inequality for eu...
3
cards
3. Isometries.
Theorem 1the euclidean metric is ...,
Definition 2distance preserving m...,
Definition 3euclidean space
6
cards
4. Colinearity
Proposition 1x y z are colinear iff,
Proposition 2motions map lines to,
Definition 7linear map
5
cards
5. Isometries as Linear Maps
Proposition 3equivalnces to t affine,
Corollary 1every motion is what,
Lemma 6polarisation identity for ...
5
cards
6. Motion Group
Definitionrotation around originr...,
Theorem 3types of motions in e2,
Theorem 4every motion in e2 is ei...
5
cards
7. Complex Linear Algebra
Definitionsesquilinear inner product,
Lemma 8extension of linear isomet...,
Definitiondirect sum
6
cards
8. Normal Form Theorem
Lemma 10if lambda is an eigenvalu...,
Theoremnormal form theorem
2
cards
9. Reflections
Definitionhyperplane,
Definition 12set of fixed points,
Definition 13reflection
13
cards
10. Spherical Geometry
Definition 15n dimensional sphere...,
Definition 16spherical line great...,
Definition 17antipodal
6
cards
11. Comparisons of Lines in R3 and S2
Proposition 8if c and d are disti...
1
cards
12. Main Formula for Spherical Geometry
Proposition 9main formula for sph...,
Lemma 16the spherical distance sa...
2
cards
13. Spherical Isometries
Lemma 17the restriction of a line...,
Lemma 18a bijective map which pre...,
Proposition 11if t is an isometry...
4
cards
14. Spherical Triangles
Spherical area assumptions,
Theorem 8girard s theorem,
Corollary 5the sum of the angles ...
3
cards
15. Hyperbolic Geometry
Definition 21lorentz norm,
Definition 20lorentz inner product,
Definition 22n dimensional hyperb...
7
cards
16. Lorentz Group
Definition 25lorentz transformati...,
Lemma 21if t is a positive biject...,
Definition 26lorentz orthogonal
4
cards
17. Hyperbolic Lines
Definition 28lorentz planehyperbo...,
Lemma 23for distinct points in hn,
Definition 29v rv is
5
cards
19. Intersection of Planes with H^2
Definitionintersection of planes ...,
Definitionhyperbolic isometries o...,
Definition 31lorentz orthogonallo...
4
cards
20. Sketch of Isomorphism between Isom and O+
Proposition 12if t is a lorentz t...,
Corollary 7a lorentz transformati...,
Lemma 25if a in o 1 n then
5
cards
21. Triangle Inequality for H^2
Definition 32hyperbolic triangle,
Definition 33lorentz cross product,
Lemma 27binet cauchy identity
6
cards
22. Projective Space
Definition 34algebraic projective...,
Lemma 30there is a bijective from...
2
cards
23. The Projective Plane
Representation of p 2 in r 3,
Definition 35k dimensional projec...,
Definition 36classification of p ...
5
cards
24. Projective Transformations and Frame of Reference
Definition 38perspectivity,
Definition 39real projective line...,
Definitin 40projective transforma...
6
cards
25. Projective Dimension Formula
Theorem 11for two projective line...,
Theorem 12any two distinct lines ...
2
cards
26. Projective Frame of Reference and Linear Maps
Theorem 13there is a bijection be...
1
cards
27. Cross Ratio
Any three points on p 1 can be ma...,
Definition 44cross ratio,
Proposition 14formula for cross r...
3
cards
28. Pappus' Hexagon Theorem
Lemma 31projective linear maps pr...,
Theorem 14pappus theorem
2
cards
29. Axiomatic Projective Geometry
Definition 45axiomatic projective...,
Proposition 15p 2 f_q satisfies
2
cards
30. Finite Axiomatic Projective Geometry
Proposition definition 16a finite...,
Definition 46two axiomatic projec...
2
cards
Exercises and Exams
11 a find matrix corresponding to...,
11 b find a b such that t x ax b ...,
11 e is the composition
34
cards
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MA243 Geometry
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