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Hyperbolic Geometry
SHIVANI BAISIWALA
SO RI BAEK
INTRODUCTION
What is it?
 Hyperbolic Geometry is form of non-
Euclidean geometry
 Obtained by considering only the first
four postulates of Euclid and replacing
the parallel postulate by its negation,
the hyperbolic axiom.
Postulates of Euclidean Geometry
1.
2.
3.
4.
5.
A straight line may be drawn from any given point to any other
A straight line may be extended to any finite length
A circle may be described with any given point as its center and
any distance as its radius
All right angles are congruent
If a straight line intersects two other straight lines, and
so makes the two interior angles on one side of it
together less than two right angles, then the other
straight lines will meet at a point if extended far enough
on the side on which the angles are less than two right
angles.
REPLACED WITH
There exists a line and an external point not lying on this
line such that there are at least two lines passing through
this point and parallel to this line – aka Hyperbolic
Axiom
Basic Properties of Hyperbolic Geometry
 All triangles have angle sum less than 180.

180 minus the angle sum of a triangle is a positive number known
as the defect of the triangle.
 Rectangles do not exist.
 If two triangles are similar, they are congruent.
 All convex quadrilaterals have angle sum less than 360.
 There are an infinite number of parallels for any given
line.
 If two angles are similar, they are also congruent.

It is impossible to magnify or shrink a triangle without distortion,
so solely similar triangles do not exist.
Viewing Hyperbolic Space
 Hyperbolic space is defined by negative curvature in
all directions.
 In a sense, it can be thought of as the opposite of a
sphere.
Models Of Hyperbolic Geometry
 Two major models – Two ways to
mathematically “view” hyperbolic
space.
Beltrami
– Klein Model
Poincare Model
Beltrami – Klein Model
 Think of a circle Y in the Euclidean plane.
 If O is the center of circle Y, and OR is a radius, the
interior of Y consists of all points X such at OX < OR
 All these points represent the points of the
hyperbolic plane in Klein’s model.
Klein Model Continued
 Remember that a chord is a segment from one end of
the circle to another.
 Now, consider that chord without its endpoints.
 In Klein’s model, these are the lines of the hyperbolic
plane.
Parallels in Klein Model
 The definition of two parallel lines is that they never
meet in the plane under discussion.
 The fact that they may meet outside the circle is
irrelevant, because that is not part of the Klein
model.
 Therefore, the two intersecting lines
are both parallel to the line at the
bottom.
The Poincare Model
 This model is a disk model that also represents the
points of the hyperbolic plane by the points interior
to a Euclidean circle Y
 Lines are defined differently.
Poincare Model Continued
 Only open diameters (chords that pass through the
center) are considered lines.
 The other lines are represented by open arcs or
circles orthogonal to Y.
 All arcs are considered parallel
as long as they don’t meet.
GEOMETRY IN THE
HYPERBOLIC PLANE
HYPERBOLIC TRIANGLES
TRADITIONAL AREA CALCULATIONS
 In traditional geometry, we calculate
areas of triangles by essentially
breaking up the area of a rectangle
into two.
 Hence, we have our traditional triangle area formula:
 However, this cannot work in hyperbolic geometry.
 Why? Because rectangles do not exist in hyperbolic
geometry, so the whole basis of our area formula is
removed.
CALCULATING AREA AND DEFECT
 In hyperbolic geometry, the area function of a triangle
has the following properties:


Invariance under congruence: Congruent triangles have the
same area.
Additivity: If a polygonT is split into two polygons – T1 and T2
– by a segment joining a vertex to a point of the opposite side,
then the area of T is the sum of the areas of T1 and T2.
 Defect is important
 This is the difference between the expected angle sum of a
given polygon in Euclidean space and its actual angle sum in
Hyperbolic space.
 It is always positive and is given by
where n is the
number of sides of the polygon.
AREA FORMULAS
 Area formula for hyperbolic geometry for a polygon
angles measured in degrees (so your answer
will end up in radians)
where is a proportionality constant that is typically
set at k = 1.
SOMETHING INTERESTING
 Since defect measures how much the angle sum is
less than 180, the highest it can be is 180 (if the
angle sum of the actual triangle is 0).
 Therefore, the maximum area of any triangle is
.
 However, no triangle actually achieve this. It is
essentially an asymptotic value, since we can
approach it as closely as we like by making the shape
of our triangle achieve a trebly asymptotic
appearance.
CACULAT E THE AREA
 Given a triangle with angle measures A = 14.2, B =
13.4, and C = 43.4, calculate the defect (in degrees)
and the area (in radians) of the triangle.
CALCULATE
 Given a triangle with area
, and two angle
measures of 32 and 40, calculate the third angle
measure of the triangle.
AREA AND CIRCUMFERENCE
 The circumference of a circle of radius
 The area of a circle with radius
is given by:
is
 The area of a hyperbolic circle is much larger than
the area of a Euclidean circle with the same radius.
CALCULATE
 Calculate the area and the circumference of a circle
with radius = 2ln2
(Hint: Recall that
)
CALCULATE
 Given a circle with radius 2, calculate the
circumference.
TESSELLATIONS
 A regular tessellation is covering of the plane by regular
polygons so that the same number of polygons meet at
each vertex.
 There are infinitely many hyperbolic tessellations..
 How do you know if something is a tessellation?

Tessellations are written in the notation {n,k}, known as the Schlafli
symbol, where n is the number of sides on the polygon and k is the
number of polygons that meet at each vertex.

If

If it is less than .5, you have a hyperbolic tessellation.
If it is greater than .5, you have an elliptic tessellation.

is equal to .5, you have a Euclidean tessellation.
EXAMPLES
HYPERBOLIC TESSELLATIONS
 You can calculate area of the shapes in hyperbolic
tessellations using the Schlafli symbol.
 Keep in mind that tessellations use regular polygons,
where all the angles are equal
 Given a tessellation {n,k},
you know that each angle
measure of the polygon used
will be
.
 Once you have the angles,
you can calculate defect, and
then calculate area.
CALCULATE
 Calculate the area of each polygon used in the
hyperbolic tessellation given by {3,10}.
CALCULATE
 Given that the area of a decagon used in a hyperbolic
tessellation is
, find the Schlafli symbol.
CREATE YOUR OWN TESSELLATION