(i) (u) Define Hyperbolic Geometry and the hyperbolic parallel postulate (b) Prove that in a...

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(i) (u) Define Hyperbolic Geometry and the hyperbolic parallel postulate (b) Prove that in a hyperbolic geometry there are no noncongruent similar triangles. $i,1iYr ' t- (c) Show that in elliptic geometry there are no similar noncongruent triangleS (Z) (u) Show that in hyperbolic geometry the common perpendicular between 2 parallel lines is the shortest distance between the lines. (b) Let L artd M be parallel lines in hyperbolic geometry. Show that there is a unique common perpendicualr. You can assume that there is at least one. f (3) (") Define defect of triangle in hyperbolic geometry / (U) Prove: Defect is additive on triangles t (") How do we measure the area of a triangle in hyperbolic geometry (d) In hyperbolic geoemtry must congruent triangles have the same area? why? ./ '/ (a) (") Describe the Klein Disk Model and the Poincare Disk Model for hyperbolic geometry. For each describe points, lines, distance and angles (b) In the Klein model why can't ordinary Euclidean angle be used? (c) In the Klein model why is the cross-ratio used to define distance? (d) In the Klein model why is the logarithm of cross-ratio used to define distance rather than just the cross ratio? (e) L and B are two points inside the unit disk with Euclidean coordinates ,q,: (i,i),A : (*, i). In the Klein model what is the hyperbolic distance between them? (5) Show that on a sphere any 2 nonantipodal points fall on a unique great circle. (0) (") Describe the real projective plane (What are points and lines) (b) Prove that in the real projective plane each two distinct points defines a unique line.
	
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PROJECT 
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(i) (u) Define Hyperbolic Geometry and the hyperbolic parallel postulate
(b) hyperbolic geometry 
Prove that in a there are no noncongruent similar
1
'
triangles. 
$i,1iYr 
t-
(c) geometry 
Show that in elliptic there are no similar noncongruent triangleS
( 
,i
(Z) (u) Show that in hyperbolic geometry the common perpendicular between 2
parallel lines is the shortest distance between the lines.
parallel 
(b) Let L artd M be lines in hyperbolic geometry. that there is
Show 
perpendicualr. You 
a unique common can assume that there is at least one.
(3) (") Define defect of triangle in hyperbolic geometry
f 
/
(,, 
I
/ (U) Prove: Defect is additive on triangles
t 
(") How do we measure the area of a triangle in hyperbolic geometry
(d) In hyperbolic geoemtry must congruent triangles have 
the same area? why?
'/
./ 
(a) (") 
x Describe the Klein Disk Model and the Poincare Disk Model for
hyperbolic geometry. For each describe points, lines, distance 
and angles
(b) 
In the Klein model why can't ordinary Euclidean angle be used?
(c) In the Klein model why is the cross-ratio used to 
define distance?
(d) In the Klein model is 
why the logarithm of cross-ratio used to define
just 
distance rather than the cross ratio?
(e) points 
L and B are two inside the unit disk with Euclidean coordinates
: 
,q,: (i,i),A (*, In the Klein model what is 
the hyperbolic distance between
i). 
them?
(5) that on a sphere any 2 nonantipodal points great 
Show fall on a unique circle.
(0) (") projective plane 
Describe the real (What are points and lines)
(b) Prove that in the real projective plane each two points 
distinct defines a
unique line.
(Z) (u) When is an axiom system consistent
(b) The fact that Klein 
the and Poincare models use Euclidean geometry
to model hyperbolic geometry says what about the consistency 
of Euclidean and
geometry.
hyperbolic 
(c) What did Beltrami do and what does this say what 
about the consistency
and hyperbolic geometry.
of Euclidean...

Robert · Accepted Answer

1) a) Hyperbolic geometry is neutral geometry together with the hyperbolic parallel postulate
 The hyperbolic parallel postulate is that given line L and P not on L then there exists at least two lines 
through P parallel to L.  
b) Lets assume ΔPQR and ΔP’Q’R’ are similar (they have the same angles), but are not congruent. The no 
corresponding sides are congruent (otherwise, they would be congruent, using the principle angle-side-angle). 
We may assume, without loss of generality, that PQ>P’Q’ and PR>P’R’.
As PQ>P’Q’ then there will be a point Q” on PQ and a point R” on PR such that PQ”=P’Q’ and PR” = P’R’. As the 
angles are same and sides are also same so by ASA, ΔP’Q’R’≡ ΔPQ”R”.  Hence angle(PQ”R”) = angle (Q’) and 
angle (PR”Q”) = angle (R’)  
But we also have that angle(R) = angle(R’)  and angle(Q) = angle(Q’) so angle(PQ”R”) = angle(Q) and 
angle(PR”Q”) = angle(R ).

(i) (u) Define Hyperbolic Geometry and the hyperbolic parallel postulate (b) Prove that in a hyperbolic geometry there are no noncongruent similar triangles. $i,1iYr ' t- (c) Show that in elliptic...

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