Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 06 Non-Euclidean Geometry=Curvature
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Name: Parallel axiom
Definition: Through any point \(p\) not on the line \(L\) there exists precisely one line through \(p\) that does not meet \(L\)
Page: 268, 269
Chapter: 6I
Section: 1, 2
Definition2: \(\mathrm{E}(T)=0\) for each triangle \(T\)

Name: Spherical axiom
Definition: Through any point \(p\) not on the line \(L\) there exists no line through \(p\) that does not meet \(L\)
Page: 268, 269
Chapter: 6I
Section: 1,2
Definition2: \(\mathrm{E}(T)>0\) for each triangle \(T\)

Name: Hyperbolic axiom
Definition: Through any point \(p\) not on the line \(L\) there exists at least two lines through \(p\) that does not meet \(L\)
Page: 268, 269
Chapter: 6I
Section: 1,2
Definition2: \(\mathrm{E}(T)<0\) for each triangle \(T\)

Name: The angular excess of \(T\)
Definition: the sum of angle of \(T\) minus \(\pi\)
Page: 269
Chapter: 6I
Section: 2
Variables: \(T\) a triangle
Context: A fixed non-euclidean geometry
Notation: \(\mathrm{E}(T)\)
Property: a real multiple of the angle of the triangle

Name: The area of \(T\)
Page: 270
Chapter: 6I
Section: 2
Variables: \(T\) a triangle
Notation: \(\mathrm{A}(T)\)

Smaller0: \(T\) and \(T'\) are similar (have the same angle)
Greater0: equivalent (have the same size)
Page: 270
Chapter: 6I
Section: 2
Variables: \(T,T'\) triangle
Smaller to greater: iff
Prefix Greater: \(T\) and \(T'\) are
Assuming: Non Euclidean geometry
Extra: i.e. if triangles have the same 3 angles, they have the same size

Question: How to define a unit of length in Non-Euclidean geometry with parameter \(k\)?
Answer: Takes the length of an equilateral triangle with a fixed angle sum
Page: 270
Chapter: 6I
Section: 2
Variables: \(T\) triangle,
\(k<0\)
Answer2: the square root of [$]\abs{k}[/$]

Name: differential geometry
Definition: study of curved surfaces by means of ideas from calculus
Page: 271
Chapter: 6I
Section: 3

Name: Pseudosphere
Page: 271
Chapter: 6I
Section: 3

Name: A line on the pseudosphere
Definition: A geodesic
Page: 271
Chapter: 6I
Section: 3
Context: Pseudosphere

Name: Curvature
Definition: \(\mathbf n\) normal to surface at \(p\)
Page: 274
Chapter: 6I
Section: 5
Notation: \(k(p)\)
Definition2: \(\Pi\) plane containing \(\mathbf n\)
Name2: intrinsic curvature
Intuition: What the surface is like in the immediate neighbourhood of \(p\)
Definition6: the (signed) curvature at \(p\) of the curve in which \(\Pi\) intersects the surface
Name3: total curvature
Definition7: the sign depends on whether the centre of curvature is in the direction \(\mathbf n\) or \(-\mathbf n\)
Name4: Gaussian curvature