Question

Write an essay comparing Euclidean Geometry with Hyperbolic Geometry?

Answer #1

(Euclidean and Non Euclidean geometry)
Consider the following statements:
Given a line l and a point P not on the line:
There exists at least one line through P which is
perpendicular to l.
There exists at most one line through P which is
perpendicular to l.
There exists exactly one line through P which is
perpendicular to l.
Prove each statement or give a
counter-example E2 (Euclidean Plane), H2 ,
(Hyperbolic Plane)and the sphere S2 (Spherical
Plane)
( Consider...

prove that rectangles dont exist in hyperbolic geometry

Use postulates (axioms) to explain how Projective geometry
differs from Euclidean geometry

Please keep it simple for Hyperbolic Geometry (the response I
was given was great, but it was well beyond what we needed)
Illustrate with a picture showing that the following are
simply not true in Hyperbolic Geometry
c). The angle sum of any triangle is 180
d). Rectangles exist

prove that the product of a rotation and a translation is a
rotation
Euclidean Geometry and Transformatons

What methodological techniques inherited from Euclidean geometry
had to be ignored in order to apply algebra to geometry?

what role did the fifth postulate play in the development of
non-Euclidean Geometry?

Prove in hyperbolic geometry: If two Saccheri quadrilaterals
have congruent bases and congruent summits, then they are
congruent.

In topology, how do the intrinsic geometry differ among a
Euclidean plane, a flat torus, a Klein bottle, and a sphere?

hyperbolic geometry: using degree measure and k=1, show that a
triangle with area 3 can be divided into three triangles each with
area 1.

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