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Luca Kollmer
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Decks in this class (30)
1. Introduction
Definition metric metric space 1,
Lemma 1 bilinearity of the inner ...
2
cards
2. Euclidean Metric
Lemma 2 cauchy schwartz inequality 1,
Lemma 3 translation invariance of...,
Lemma 4 triangle inequality for e...
3
cards
3. Isometries.
Theorem 1 the euclidean metric is...,
Definition 2 distance preserving ...,
Definition 3 euclidean space 3
6
cards
4. Colinearity
Proposition 1 x y z are colinear ...,
Proposition 2 motions map lines to 2,
Definition 7 linear map 3
5
cards
5. Isometries as Linear Maps
Proposition 3 equivalnces to t af...,
Corollary 1 every motion is what 2,
Lemma 6 polarisation identity for...
5
cards
6. Motion Group
Definition rotation around origin...,
Theorem 3 t ax b motion 2,
Theorem 4 every motion in e2 is e...
5
cards
7. Complex Linear Algebra
Definition sesquilinear inner pro...,
Lemma 8 extension of linear isome...,
Definition direct sum 3
6
cards
8. Normal Form Theorem
Lemma 10 if lambda is an eigenval...,
Theorem normal form theorem 2
2
cards
9. Reflections
Definition 11 hyperplane 1,
Definition 12 set of fixed points 2,
Definition 13 reflection 3
13
cards
10. Spherical Geometry
Definition 15 n dimensional spher...,
Definition 16 spherical line grea...,
Definition 17 antipodal 3
6
cards
11. Comparisons of Lines in R3 and S2
Proposition 8 if c and d are dist...
1
cards
12. Main Formula for Spherical Geometry
Proposition 9 main formula for sp...,
Lemma 16 the spherical distance s...
2
cards
13. Spherical Isometries
Lemma 17 the restriction of a lin...,
Lemma 18 a bijective map which pr...,
Proposition 11 if t is an isometr...
4
cards
14. Spherical Triangles
Spherical area assumptions 1,
Theorem 8 girard s theorem 2,
Corollary 5 the sum of the angles...
3
cards
15. Hyperbolic Geometry
Definition 21 lorentz norm 1,
Definition 20 lorentz inner produ...,
Definition 22 n dimensional hyper...
7
cards
16. Lorentz Group
Definition 25 lorentz transformat...,
Lemma 21 if t is a positive bijec...,
Definition 26 lorentz orthogonal 2
4
cards
17. Hyperbolic Lines
Definition 28 lorentz plane hyper...,
Lemma 23 for distinct points in hn 2,
Definition 29 v rv is 3
5
cards
19. Intersection of Planes with H^2
Definition intersection of planes...,
Definition hyperbolic isometries ...,
Definition 31 lorentz orthogonal ...
4
cards
20. Sketch of Isomorphism between Isom and O+
Proposition 12 if t is a lorentz ...,
Corollary 7 a lorentz transformat...,
Lemma 25 if a in o 1 n then 3
5
cards
21. Triangle Inequality for H^2
Definition 32 hyperbolic triangle 1,
Definition 33 lorentz cross produ...,
Lemma 27 binet cauchy identity 3
6
cards
22. Projective Space
Definition 34 algebraic projectiv...,
Lemma 30 there is a bijective fro...
2
cards
23. The Projective Plane
Representation of p 2 in r 3 1,
Definition 35 k dimensional proje...,
Definition 36 classification of p...
5
cards
24. Projective Transformations and Frame of Reference
Definition 38 perspectivity 1,
Definition 39 real projective lin...,
Definitin 40 projective transform...
6
cards
25. Projective Dimension Formula
Theorem 11 for two projective lin...,
Theorem 12 any two distinct lines...
2
cards
26. Projective Frame of Reference and Linear Maps
Theorem 13 there is a bijection b...
1
cards
27. Cross Ratio
Any three points on p 1 can be ma...,
Definition 44 cross ratio 2,
Proposition 14 formula for cross ...
3
cards
28. Pappus' Hexagon Theorem
Lemma 31 projective linear maps p...,
Theorem 14 pappus theorem 2
2
cards
29. Axiomatic Projective Geometry
Definition 45 axiomatic projectiv...,
Proposition 15 p 2 f_q satisfies 2
2
cards
30. Finite Axiomatic Projective Geometry
Proposition definition 16 1,
Definition 46 two axiomatic proje...
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 of two re...
34
cards
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MA243 Geometry
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