MA243 Geometry

This class was created by Brainscape user Luca Kollmer. Visit their profile to learn more about the creator.

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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