Other Geometries
There are MANY geometries other than the familiar Euclidean. We will look at several of them.
Please pay attention to what is similar and what is different about each as we progress.
Finite Geometries
Incidence Geometry
Taxicab Geometry
Spherical Geometry
Hyperbolic Geometry
1
Introduction to Axiomatic Systems
In studying any geometry, it is important to note the axiomatic framework of the geometry and
keep it in mind. Often students are so challenged by the details that they forget that there is a
structure to geometry. Each geometry has a framework called its axiomatic system. An outline of
a typical axiomatic system is below.
Any axiomatic system has four parts:
undefined terms
axioms (also called postulates)
definitions
theorems
The undefined terms are a short list of nouns and relationships. These terms may be visualized
but cannot be defined. Any attempt at a definition ends up circling around the terms and using
one to define the other. These are the basic building blocks of the geometry. It is usually a good
idea to have a mental image of the undefined terms – a visualization of the objects and how they
relate.
Axioms (or postulates) are a list of rules that define the basic relationships among the undefined
terms and make clear the fundamentals facts about a system. Axioms are always true for the
system. No deviation from the facts they state is permitted in working with the system.
Definitions and theorems build on the axioms and undefined terms, clarifying relationships and
auxiliary facts.
We will be using, with slight modification, the set of undefined terms and axioms developed by
The School Mathematics Study Group during the 1960’s for this module. This list of axioms is
not as brief as one that would be used by graduate students in a mathematics program nor as long
as some of those systems in use in middle school textbooks. One definite advantage to the
SMSG list is that it is public domain by design. We will be using the Cartesian coordinate plane
as our visualization of the undefined terms of Euclidean geometry.
Once we have spent time learning the axioms, some definitions and a few theorems we will
move to the second module on Euclidean Topics and look at geometric shapes and proofs that
require using the axioms, definitions, and theorems in concert
2
In the following geometries:
In the following two examples of finite geometries, each has exactly one model and neither has
an alternative model with more or fewer points. The axioms are quite specific and controlling on
this issue.
Note that the axioms are quite specific about which undefined terms are “incident” or bearing
upon one another in all three geometries.
Then we will explore another type of geometry is called an Incidence Geometry.
The axioms for an Incidence Geometry are specific about a couple of things but do allow at least
two distinctly different models.
In TCG, EG, SG there are only one model. HG has several 2D and a 3D model!
In Euclidean Geometry, there is exactly one line through a given point not on a given line that is
parallel to the given line. In the following geometries, some have NO parallel lines, others have
more than one through that point. It’s a variable kind of thing!
3
The Three Point Geometry
Undefined terms:
point, line, on
Axioms:
A1
A2
A3
A4
There are exactly three distinct points.
Two distinct points are on exactly one line.
Not all the points are on the same line.
Each pair of distinct lines is on at least one point.
Model
Three Point Geometry exercise:
There’s only one basic model for this geometry. Sketch it here:
Possible Definitions:
Collinear Points – points that are on the same line are collinear.
Theorems:
Theorem 1:
Each pair of distinct lines is on exactly one point.
Theorem 2:
There are exactly 3 distinct lines in this geometry.
4
Proof of Theorem 1
Theorem 1:
Each pair of distinct lines is on exactly one point.
Suppose there’s a pair of lines on more than one point. This cannot be because then the two lines
have at least two distinct points on each of them and Axiom 2 states that
A2
Two distinct points are on exactly one line.
Thus our supposition cannot be and the theorem is true.
QED
This type of proof is called a proof by contradiction. It works like a conversation.
Someone asserts something and someone disagrees and contradicts them. The assertion is the
theorem and the contradiction is the sentence that begins with “Suppose…”.
Then the first person points out why the supposition cannot possibly be true…which has the
handy property that it proves the theorem.
The proper contradiction to an assertion that “exactly one” situation is true is to suppose that
“more than one” is true.
Note: we have no need of a distance formula or a way to measure angles with this simple little
geometry. These things MUST be defined in the axioms for you to have them!
Given a line and a point not on that line, how many lines are there through that point and parallel
to the given line?
5
The Five Point Geometry
Undefined terms
Axioms
A1
A2
A3
point, line, on
There are exactly five points.
Any two distinct points have exactly one line on both of them.
Each line is on exactly two points.
Models
P1
Points: {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 , 𝑃5 }
P5
P2
P4
P3
̅̅̅̅̅̅
̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅
Lines: {𝑃
1 𝑃2 , 𝑃1 𝑃3 , 𝑃1 𝑃4 , 𝑃1 𝑃5 , 𝑃2 𝑃3 , 𝑃2 𝑃4 , 𝑃2 𝑃5 , 𝑃3 𝑃4 , 𝑃3 𝑃5 , 𝑃4 𝑃5 }
Note that the lines crossover one another in the interior of the
“polygon” but DO NOT intersect at points. There are only 5
points! LINE STUFF!
Possible Definitions
Triangle -- a closed figure formed by 3 lines. An example: 𝑃1 𝑃2 𝑃3 is a triangle.
How many triangles are there?
Quadrilateral – a closed figure formed by 4 lines. An example: 𝑃2 𝑃5 𝑃4 𝑃3 is a quadrilateral.
How many quadrilaterals are there?
Parallel lines – two lines are parallel if they share no points.
6
̅̅̅̅̅̅
̅̅̅̅̅̅
̅̅̅̅̅̅
Note that ̅̅̅̅̅̅
𝑃1 𝑃2 is parallel to line𝑃
4 𝑃5 . So are 𝑃3 𝑃4 and 𝑃2 𝑃5 …
Let’s look at a given line and a given point…
Other possible definitions:
Collinear points
Interior
Plane
Theorem
Each point is on exactly 4 lines.
7
An Incidence Geometry
Undefined terms:
point, line, on
Axioms:
A1
A2
A3
A4
There is exactly one line on any two distinct points.
Each line has at least two distinct points on it.
There are at least three points.
Not all the points lie on the same line.
Note: no distance, no angle measure
Models:
Two examples follow; there are others.
Definitions – We’ll look at the models and see what makes sense…
Parallel lines
The distance from point one to point two
Intersecting lines
Triangles
Quadrilaterals
Between or interior
Concurrent lines – share a point…more than 2 are “concurrent”
Theorems:
Theorem 1:
Theorem 2:
Theorem 3:
If two distinct lines intersect, then the intersection is exactly one point.
Each point is on at least two lines.
There is a triple of lines that do not share a common point.
Theorems must be totally true in EVERY MODEL!
8
An Incidence Geometry, continued
A six point model:
B
The ONLY points are the 6 dots that are labeled. Note
that in the interior of the “polygon” there are NO
intersections of lines at points.
C
A
D
F
E
AB
AC
AD
AE
AF
BC
BD
BE
BF
A1
A2
A3
A4
There is exactly one line on any two distinct points. See the list
Each line has at least two distinct points on it. See the “endpoints”.
There are at least three points. There are 6 which is “at least 3”.
Not all the points lie on the same line. See the list.
Theorem 1:
Theorem 2:
Theorem 3:
CD
CE
CF
DE
DF
Imagine the points are little Styrofoam balls and that
the lines are pipe cleaners…where two pipe cleaners
lay on top of each other there’s no intersection only a
“crossover”. Only at the ends where the ends are stuck
into the balls is there a point and an intersection.
The points are: A, B, C, D, E, and F.
The lines are:
EF
If two distinct lines intersect, then the intersection is exactly one point.
Each point is on at least two lines.
There is a triple of lines that do not share a common point.
Theorem 1:
For example: lines BF and BE intersect only at B.
The “crossovers” in the interior are not intersections.
Theorem 2:
Each point is on 5 lines which is “at least 2”.
Theorem 3:
All you have to do with Theorem 3 is show one triple:
AB, CF, and ED do not share a common point.
9
Let’s look at the situation with respect to parallel lines.
We will use the definition that two lines parallel lines if they share no points.
In Euclidean Geometry, if you have a line and a point not on that line, there is exactly one line
through the point that is parallel to the given line.
Let’s check this out:
Take line AC and point B. These are a line and a point not on that line.
B
C
A
D
F
E
Now look at lines BF, BE, and BD. Both of these lines are parallel to line AC.
(recall that the lines that overlap in the “interior of the pentagon” do NOT intersect at a point –
there are only 6 points in this geometry).
So there are exactly THREE lines parallel to a given line that are through a point not on the
given line. This is certainly non-euclidean!
10
An Incidence Geometry, continued
A Model with an infinite number of points and lines:
Points will be {(𝑥, 𝑦)|𝑥 2 + 𝑦 2 < 1}, the interior of the Unit Circle, and lines will be the set of all
lines that intersect the interior of this circle.
So our model is a proper subset of the Euclidean Plane.
Model:
Note that the labeled points (except H) are NOT
points in the geometry. A is on the circle not an
interior point. It is convenient to use it, though.
A
B
H
P1
H is a point in the circle’s interior and IS a point
in the geometry.
P2
C
D
G
F
We cannot list the number of lines – there are an
infinite number of them.
E
Checking the axioms:
A1
A2
A3
A4
There is exactly one line on any two distinct points.
This model is a subset of Euclidean geometry and the axiom holds.
Each line has at least two distinct points on it.
Each line has an infinite number of points by Euclidean Axioms.
There are at least three points.
The unit disc has an inifinite number of points.
Not all the points lie on the same line.
True
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Definitions:
Parallel lines: lines that share no points are parallel.
In Euclidean Geometry, there is exactly one line through a given point not on a given line that is
parallel to the given line.
Interestingly, in this geometry there are more than two lines through a given point that are
parallel to a given line.
Let’s look at lines GC and GB. They intersect at
G…which is NOT a point in the geometry. So
GC and GB are parallel. In fact, they are what is
called asymptotically parallel. They really do
share no points.
A
B
H
P1
P2
C
D
G
F
E
Now look at P1P2. It, too, is parallel to GC.
Furthermore both P1P2 and GB pass through
point H.
P1P2 is divergently parallel to GC.
Not only is the situation vis a vis parallel lines different, we even have flavors of parallel:
asymptotic and divergent. So we are truly non-euclidean here, folks.
Theorem 1:
Theorem 2:
Theorem 3:
If two distinct lines intersect, then the intersection is exactly one point.
Inherited from Euclidean Geometry.
Each point is on at least two lines.
Each point is on an infinite number of lines.
There is a triple of lines that do not share a common point.
FE, GC, and AD for example.
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The SMSG Axioms for Euclidean Geometry
Undefined Terms:
point, line, and plane
We take as our beginning point the undefined terms:
point, line, and plane.
Most people visualize a point as a tiny, tiny dot. Lines are thought of as long, seamless
concatenations of points and planes are composed of finely interwoven lines: smooth, endless
and flat.
Think of undefined terms as the basic sounds in a language – the sounds that make up our
language for the most part have no meaning in themselves but are combined to make words.
The grammar of our language and a good dictionary are what make the meaning of the sounds.
This part of language corresponds to the axioms and definitions that you will find next in the
module.
From there the facts, flights of fancy, and content-laden sentences are built – these are the
theorems and definition in an axiomatic system.
The conventions of the Cartesian plane are well suited to assisting in visualizing Euclidean
geometry. However there are some differences between a geometric approach to points on a line
and an algebraic one, as we will see in the explanation of Axiom 3.
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SMSG Postulates for Euclidean Geometry:
A1.
Given any two distinct points there is exactly one line that contains them.
A2.
The Distance Postulate: To every pair of distinct points there corresponds a unique
positive number. This number is called the distance between the two points.
A3.
The Ruler Postulate: The points of a line can be placed in a correspondence with the
real numbers such that
A.
To every point of the line there corresponds exactly one real number.
B.
To every real number there corresponds exactly one point of the line,
and
C.
The distance between two distinct points is the absolute value of the difference of
the corresponding real numbers.
A4.
The Ruler Placement Postulate: Given two points P and Q of a line, the coordinate
system can be chosen in such a way that the coordinate of P is zero and the coordinate of
Q is positive.
A5.
A.
B.
A6.
If two points line in a plane, then the line containing these points lies in the same
plane.
A7.
Any three points lie in at least one plane, and any three non-collinear points lie in
exactly one plane.
A8.
If two planes intersect, then that intersection is a line.
A9.
The Plane Separation Postulate: Given a line and a plane containing it, the points of the
plane that do not lie on the line form two sets such that
A.
each of the sets is convex, and
B. if P is in one set and Q is in the other, then segment PQ intersects the line.
A10.
The Space Separation Postulate: The points of space that do not line in a given plane
form two sets such that
A.
each of the sets is convex, and
B. if P is in one set and Q is in the other, then the segment PQ intersects
the plane.
A11.
The Angle Measurement Postulate: To every angle there corresponds a real number
between 0 and 180.
Every plane contains at least three non-collinear points.
Space contains at least four non-coplanar points.
14
A12.
The Angle Construction Postulate: Let ⃗⃗⃗⃗⃗
𝐴𝐵 be a ray on the edge of the half-plane 𝐻.
For every 𝑟 between 0 and 180 there is exactly one ray ⃗⃗⃗⃗⃗
𝐴𝑃 with 𝑃 in 𝐻 such that
𝑚∠𝑃𝐴𝐵 = 𝑟.
A13.
The Angle Addition Postulate: If 𝐷 is a point in the interior of ∠𝐵𝐴𝐶, then 𝑚∠𝐵𝐴𝐶 =
𝑚∠𝐵𝐴𝐷 + 𝑚∠𝐷𝐴𝐶.
A14. The Supplement Postulate: If two angles form a linear pair, then they are
supplementary
A15.
The SAS Postulate: Given an one-to-one correspondence between two triangles (or
between a triangle and itself). If two sides and the included angle of the first triangle are
congruent to the corresponding parts of the second triangle, then the correspondence is a
congruence.
A16
The Parallel Postulate: Through a given external point there is at most one line parallel
to a given line.
A17.
To every polygonal region there corresponds a unique positive number called its
area.
A18.
If two triangles are congruent, then the triangular regions have the same area.
A19.
Suppose that the region R is the union of two regions R1 and R2. If R1 and R2
intersect at most in a finite number of segments and points, then the area of R is the
sum of the areas of R1 and R2.
A20.
The area of a rectangle is the product of the length of its base and the length of its
altitude.
A21.
The volume of a rectangular parallelpiped is equal to the product of the length of its
altitude and the area of its base.
A22.
Cavalieri’s Principal: Given two solids and a plane. If for every plane that intersects the
solids and is parallel to the given plane, the two intersections determine regions that have
the same area, then the two solids have the same volume.
15
Taxicab Geometry

uses the SAME undefined terms and axioms as EG up to SAS A15.

is based on the Cartesian Plane. We will use the SAME points in the plane as we use for
Euclidean Geometry and the same lines. We will use the same way of measuring angles,
too. BUT we will use a DIFFERENT formula to measure distance.
Let’s first review the old distance on a line formula: |𝑎 − 𝑏| in one dimension and in two
dimensions: √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Where Euclidean Geometry then uses the Pythagorean Theorem to get distance, we will simple
ADD the two distances:
Do you see WHY it’s called “Taxicab”?
16
HANDOUT 6.1 A
Let’s find some Taxicab distances and some Euclidean distances and see what’s going on.
Find the distances between the following points:
Points
Distance E
Distance T
(0, 0) to (0, 4)
(1, 1) to (3, 4)
(−1, 2) to ( −5, −2)
(0, 0) to (2, 2)
(0, 0) to (5, 0)
What do you notice?
Another interesting point to note is that the Euclidean distance is always less than or equal to the
Taxicab distance (p. 211).
Let’s explore a triangle or two. BOTH geometries have triangles, but Taxicab Geometry has
some surprises in theirs.
17
HANDOUT 6.1 B
Let’s study some triangles. Grab a piece of graph paper and get started:
Here’s one:
Find the vertices (1, 2), (3, 2) and (2, 3). Now let’s list the Euclidean characteristics.
Side lengths (Euclidean):
Angle measurements:
Now those angle measurement stay the same in TCG.
Side lengths (TCG):
Oops what did we find?
Here’s another:
(5, 4), (5, 1) and (9, 1)
It’s a right triangle in both geometries, isn’t it?
Let’s measure the sides.
EG:
TCG:
Ooops! What theorem is NOT true in TCG?
18
Here’s a third one:
Let’s move into Quadrant 2: Put on the base: (3, −5) and (3, −8). Let’s use TRIG to find the
coordinates of the apex angle:
Recall:
cos(30°) =
𝑎𝑑𝑗 √3
=
ℎ𝑦𝑝
2
we can solve for the height with this!
Now, let’s see if it’s equilateral (EG)…and what about those angle measures, then?
Now what about in TCG?
Back to the section:
Now let’s review congruence in the Big picture:
For two figures to be congruent, corresponding sides and corresponding angles need to be
congruent. Let’s note that the perimeter and the area of congruent triangles have the same
measure.
Now, in reality, that is SEVEN things to check:
First that you have a reasonable correspondence,
Then 3 angles and 3 sides.
But we all know the Euclidean shortcut: SAS.
Would it surprise you to know that the shortcut does NOT always work in TCG?
Let’s look at two situations: one where it does work and one where it doesn’t and see if we can
figure out a way to predict when it’s safe to use the shortcut.
19
HANDOUT 6.1 B
Here’s a situation when it DOES work:
We’ll take a 3-4-7 right triangle (TCG; 3-4-5 EG, of course) and we’ll go 𝐹(𝑥, 𝑦) = 𝐹(−𝑥, 𝑦 − 7)
[let’s translate that to English before we start]
Start with (5, 1), (9, 1) and (5, 4). Now transform it.
Ok. Are they congruent and does SAS work?
Now let’s take this pair:
(1, 2), (3, 2) and (2, 3) and let’s use the 2 – 90 degrees – 2 as our SAS
Compare this to
(−3, 0), (−1, 0) and (−1, 2)
And we’ll use the 2 – 90 degrees – 2 as our SAS. Are THESE two congruent?
Now let’s get to the bottom of why the first one works and the second one doesn’t.
Would a 45 degree rotation about the origin make the first triangle and the rotated triangle
congruent in TCG? Would a translation of 6 cm along a line at 30 degrees to the x-axis work?
What was so special about that glide reflection I chose? Is it that it was a glide reflection or
something else?
20
Now let’s continue looking at things that are the same and different in TCG and EG.
Circles! Both geometries have circles.
Let’s review Euclidean circles:
Definition:
The set of points that are equidistant from a given center. The distance from the
center to a point in the set is called the radius.
HANDOUT 6.1 C
Center: (0,0)
Radius 4.
Grab some graph paper! Let’s graph a TCG circle with the above facts.
HANDOUT 6.1 D
Now let’s look at intersections and tangencies. Euclidean circles can share:
No points
One point (internal and external)
Two points.
Taxicab circles can share
No points,
One point
Two points
An infinite number of points…let’s sketch some of these!
So, now there are some similarities and differences between TCG and EG. Let’s list a few:
21
The undefined terms and the first 14 axioms are the same. A15 is not true for TCG! Let’s make
a list and have a discussion!
Points, the Cartesian Plane…these are the same
Distance formulas are different; angle measuring is the same
Triangles
Congruence
Circles
Now let’s do an exercise with a perpendicular bisector.
Definition:
the line that is equidistant from the endpoints of a given segment
HANDOUT 6.1 E
Using the line segment (2, 3) and (6, 5). Find the midpoint of this line. Is the midpoint on the
bisector?
Using the taxicab distance find 3 points above and 3 points below the line that are equidistant
from the endpoints.
Now find 3 more above and below. Did something unusual happen?
What’s the same and what’s different here?
22
Let’s talk about trigonometry. If we use a 3 – 4 – 5 Euclidean right triangle we know things
about the angles of that triangle. We measure the angles in TCG with the same Euclidean
Protractor that we always use. Does trig hold up if we change the hypotenuse length to the TCG
required “7”?
Would it surprise you to know that the trig we learned is Euclidean and that there’s an entirely
different trig called Taxicab Trigonometry?
23
Spherical Geometry
Spherical Geometry
6.2
How, exactly, do you indicate a point on the surface of a sphere? Because it is 3D, we need
three items of information – two angles and one length.
𝜌(rho) is the length of the vector from the center of the sphere to the point. The formula is
√𝑥 2 + 𝑦 2 + 𝑧 2
𝜃theta) is the angle that the vector makes when projected into the x-y plane. It is measured
from the positive x-axis just like we do for Euclidean trigonometry work


𝜑phi, or Fe as in Fe Fi Fo Fum) is measured from the positive z-axis down to the vector.
We will stick to radian measure for 𝜑 and can use either degrees or rad for𝜃.
Let’s get a sketch* or two up: First let’s look at 𝜑 then we’ll look at 𝜃
*leave them both up!
We’ll symbolize the x-y plane as a line and show the z-axis going up. There’s 𝜑
24
Then we’ll look down straight at the z-axis into the x-y plane and see theta. Do you see “r”
The radius? It’s still √𝑥 2 + 𝑦 2 . And that’s a reason why the vector length is 𝜌.
Now let’s look at some Euclidean trig in the first drawing. Do you see the right angle with it’s
base in the x-y plane?
Note that we can use the motion principle and move the base of the angle “r” to the top to discuss
the trig of 𝜑.
sin 𝜑 =
𝑜𝑝𝑝 𝑟
= → 𝑟 = 𝜌 ⋅ sin 𝜑
ℎ𝑦𝑝 𝜌
cos 𝜑 =
𝑎𝑑𝑗 𝑧
= → 𝑧 = 𝜌 ⋅ cos 𝜑
ℎ𝑦𝑝 𝜌
Now 𝜑 is limited to 0 ≤ 𝜑 ≤ 180° and 𝜃 goes from 0 ≤ 𝜃 ≤ 2𝜋.
25
We have a similar situation in the xy-plane for theta and trig functions:
𝑥 = 𝑟 ⋅ cos 𝑡
𝑦 = 𝑟 ⋅ sin 𝑡
tan 𝑡 =
𝑦
,𝑥 ≠ 0
𝑥
Now (𝑥, 𝑦, 𝑧) coordinates can be transformed into (𝜌, 𝜃, 𝜑) in the following way:
𝑥 = 𝜌 ⋅ sin 𝜑 cos 𝜃
𝑦 = 𝜌 ⋅ sin 𝜑 cos 𝜃
𝑧 = 𝜌 ⋅ cos 𝜑
ILLUSTRATION 6.2 A
If 𝜌 = 2, 𝜃 = 60°, 𝜑 = 30° …sketch the point and show the Cartesian Coordinates.
26
Spherical Distance:
Let’s review:
Euclidean Distance
Taxicab Distance
Now: Distance in Spherical Geometry is
𝑥1 𝑥2 + 𝑦1 𝑦2 + 𝑧1 𝑧2
𝜌 ⋅ cos−1 (
)
𝜌2
Now you’ll need a calculator for the cosine inverse part of things.
Let’s practice!
Rho is 3, (0,3,0) and ( (3, 0, 0)
What is the distance between these points in SG and where are they on the sphere of radius 3?
27
So all of these geometries have distance, given in the axioms, and each uses a different formula
for it.
Latitude and Longitude:
Latitude – horizontal circles (only one, the equator, is a great circle!) and Longitude – these are
all great circles through the North and South Pole.
The latitude degrees are the complement to fe! Take 90° – fe to get the degrees latitude. Now
each degree is divided into 60 minutes and each minute is divided into 60 seconds…Babylonian
based arithmetic here!
And you measure off the equator N or S
The book uses sigma, 𝜎, for this angle measure.
28
Now let’s look at Spherical Geometry in a holistic sense.
There are triangles. Compare and contrast these triangles to Euclidean ones!
What about the “interior” and “exterior” triangles in Spherical?
How does this compare to the Cartesian Plane?
How many degrees to the interior angle sum? “excess”
How about the exterior of the triangle? Do we concern ourselves with the exterior in
Euclidean geometry? Here, we do. We can discuss the angle sum of the exterior of a triangle!
The measure of the exterior of an angle is 360° minus the measure of the interior of an angle. (in
radian measure 2𝜋!)
So the angle sum of the exterior of a triangle is adding up all the measures of the exterior angles
(and in radian measure this is 6𝜋 – the angle sum of the interior measures)
29
Example:
can we have a spherical triangle with 75, 85, and 45 degrees?
75 + 85 + 45 = 205. What is the excess? What is the measure of each exterior angle?
How are the angle sums of the interior and exterior related?
Note that you can talk about the excess of the exterior, too. It’s the angle sum of the exterior
minus pi!
What this the excess of the exterior of the 75-85-45 from above?
Do these triangles have area? Yes! And it is calculated with a formula, too. The Spherical area
of a triangle formula is based on the angle measure NOT the side lengths! The area of a
spherical geometry triangle is the triangle’s excess times rho-squared. The area of the
EXTERIOR of a spherical triangle is the excess of the exterior times rho-squared.
Take the 75-85-45 triangle from above. What is its area? What is the area of the exterior?
30
Note that if you are using a sphere with radius one, and you add the area of the interior and the
area of the exterior, you get 4pi which is the surface area of the unit sphere! P. 224
What about biangles?
Lunes or Biangles are a shape unique to Spherical geometry.
This is the simplest polygon on our surface. Build a lune on your model.
Keep in mind that the vertices are antipodal points**. Pairs that are 180° apart.
Does Euclidean geometry have antipodal points?
It is important to focus on the geometric facts that we’ve been discussing all along:
Shapes (biangle),
names for places and parts of figures (vertices),
measurements of angles (rads)
and of AREA:
𝐴𝑟𝑒𝑎𝑙𝑢𝑛𝑒 = 2𝜌2 𝛼
Where alpha is the lunar angle measure.
The summit angles are congruent in biangles
31
How about points? All the spherical points are actually points in Euclidean 3 space.
BUT SG has “Antipodal Points”**
BETWEENESS is different
Let’s sketch a great circle and talk about between. Now let’s sketch a Euclidean line and
talk about between
How about circles? Do we have them?
Well, yes and no..we need to distinguish between Great Circles (lines) and circles…
And how about that two center thing?
32
Can we have isometries in Spherical Geometry? SURE! And we use matrices to accomplish
them, too!
Let’s look at a cross section of the sphere bordered by a great circle. We can put on the circle
and the center. Now if we sketch two radii (length rho, remember?)– with a central angle of
theta and then connect the circle points, we have a CHORD, of length L.
Sketch this.
Now we can use the Law of Cosines to find the length of L (given the measure of theta), or to
find cos (theta) if given the length L.
The formula for the Law of Cosines is 𝐿2 = 𝜌2 − 2𝜌 cos 𝜃
Let’s look at how you back solve for theta.
Example:
Given 𝜌 = 4 and 𝜃 = 45°, find 𝐿
Given 𝜌 = 2 and 𝐿 = 3, find 𝜃.
33
Now, let’s remember that using an orthogonal matrix makes an isometry in 3D. And an
orthogonal matrix is special: 𝑀−1 = 𝑀𝑇 .
A reflection across the yz-plane – picture this – is F (x, y, z) = F (− x, y, z)
and the matrix is
−1 0 0
[ 0 1 0]
0 0 1
Let’s try this with (2, 1, 0)
34
A rotation about the z axis by an angle theta – picture this! Uses this matrix
cos 𝜃
[ sin 𝜃
0
− sin 𝜃
cos 𝜃
0
0
0]
1
Example:
Filling in 90° for 𝜃 will move a point from the x axis to the y axis – Let’s check this with
(3, 0, 0)
Remember that cos 90° = 0 and sin 90° = 1
(on the unit circle, (0, 1) is 90 degrees!)
Note, too, that composition of isometries doesn’t work because matrices do NOT commute!
See the example on page 228.
35
Saccheri Quadrilaterals
This particular kind of quadrilateral appears in Euclidean, Spherical, and Hyperbolic geometries.
It was discovered in the early 1700’s. Saccheri was a Jesuit priest and logician. He was a
student of Ceva, another famous geometer.
Here’s the definition:
A bi-perpendicular quadrilateral with right angles at the base and congruent legs.
Let’s sketch this.
Now, in Euclidean Geometry the summit angles are right angles. In Spherical Geometry the
summit angles are obtuse and in Hyperbolic Geometry (coming next week), the angles are acute.
36
Let’s discuss the lengths of the summit and base. EG? Let’s check on the model for SG and
what do you suppose it is for HG.
Some theorems for the Saccheri Quadrilateral that are true in all three geometries:
The diagonals are congruent.
The summit angles are congruent.
37
Hyperbolic Geometry Enrichment and Workbook
38
Historical Background of Non-Euclidean Geometry
About 575 B.C. Pythagoras wrote his book on Geometry. Some of the material was known in
other cultures centuries before he wrote it down, of course. His was the first axiomatic approach
to organizing the material. Interestingly, he did as much work as he could before introducing the
Parallel Postulate. Many people have interpreted this progression in his work as indicating a
level of discomfort with the Parallel Postulate. It’s not really possible to know what he was
really thinking. About 400 years after the birth of Christ, Proclus, a Greek philosopher and head
of Plato’s Academy, wrote a “proof” that derived the Parallel Postulate from the first 4
Postulates. thereby setting the tone for research in Geometry for the next 1400 years. Johann
Gauss, the great German mathematician, actually realized that there was another choice of axiom
but didn’t choose to publish his work for fear of getting into the same trouble as other scholars
had with the Catholic Church.
Around 1830, two young mathematicians published works on Hyperbolic Geometry –
independently of one another. The world took no note of them. In 1868, the Italian Beltrami
found the first model of Hyperbolic Geometry and in 1882, Henri Poincare developed the model
we’ll study.
120 years later, Hyperbolic Geometry is finally making it into high school textbooks. My
favorite is a text that St. Pius X used in the 90’s. If you ever get a chance to look at it – it’s just
terrific. And it includes a section on Spherical Geometry as well:
Geometry, second edition by Harold Jacobs. ISBN: 0-7167-1745-X (copyright 1987).
Note that St. Pius isn’t a flagship diocese school – they actually have a very full section of
“Algebra half”, the course for the kids not ready for Algebra I.
39
In the 1400 years of work on the axioms of Absolute Geometry there was always a special group
of people who endeavored to prove that the Parallel Postulate was actually a theorem. In fact,
Saccheri and Lambert who both came so close to realizing that there was an alternate geometry
out there waiting to be discovered never got all the way past believing that the axiom was a
theorem. Here is a list of statements that are equivalent to P1, the Euclidean Parallel Postulate:








The area of a triangle can be made arbitrarily large.
The angle sum of all triangles is a constant.
The angle sum of any triangle is 180.
Rectangles exist.
A circle can be passed through any 3 noncollinear points.
Given an interior point of a angle, a line can be drawn through that point intersecting both
sides of the angle.
Two parallel lines are everywhere equidistant.
The perpendicular distance from one of two parallel lines to the other is always bounded.
In the Hyperbolic Workbook section, I’ll ask you for illustrations from the Hyperbolic sketches
in Sketchpad that show that each of these is NOT true in Hyperbolic Geometry.
So now let’s adopt
Axiom P-2 Hyperbolic Parallel Postulate
If L is any line and P any point not on L, there exists more than one line through P parallel to L.
40
Hyperbolic Geometry Workbook
The Poincare model for Hyperbolic Geometry is the following:
Points are normal Euclidean points in the Cartesian Plane that are included in a disc:
{(𝑥, 𝑦)|𝑥 2 + 𝑦 2 < 𝑟}
[We usually pick r = 1.]
The points of the circle that encloses the disc are NOT points of Hyperbolic Geometry nor are
any points exterior to the circle.
Lines are arcs of orthogonal circles to the given circle. A circle that is orthogonal to the given
circle intersects it in two points and tangent lines to each circle at the point of intersection are
perpendicular. Note that diameters of the disc are lines in this space even though they don’t look
like arcs; each diameter is said to be an arc of a circle with a center at infinity.
Here are two orthogonal circles in a Sketchpad graph. Draw in the tangent lines to see this!
41
fx =
1-x2
1.4
gx = - 1-x2
5
hx =
8
5
qx = -
 
 
- x-
8
- x-
5
4
5
4
2
+
2
+
1
1.2
4
1
1
4
0.8
0.6
0.4
0.2
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
42
Enrichment 1:
Lines in our space:
Poincaré Disk Model
This sketch depicts the hyperbolic plane H2 usin g the Poincaré disk model. In this model, a line through
tw o poin ts is def ined as the Euclidean arc passing through the poin ts and perpendic ular to the circle.
Use this document's custom tools to perf orm constructions on the hyperbolic plane, comparing your f in dings
to equivale nt constructions on the Euclid ean plane.
Disk Controls
B
D
A
C
P. Disk Center
blue circle...not part of our space
̅̅̅̅ which is part of the Euclidean
Here is a sketch from Sketchpad that shows hyperbolic line H𝐴𝐵
circle intersecting the big blue circle (the given circle that is the space boundary). Sketch in the
̅̅̅̅. Do you see that the tangent lines are perpendicular?
tangent lines to the blue circle and H𝐴𝐵
̅̅̅̅ is a diameter of the blue circle. It, too, is a line in our space. Draw in the
Hyperbolic line 𝐴𝐵
tangents and you’ll see why.
To find this space in Sketchpad: open the Sketchpad Program files, select
Samples/Sketches/Investigations/PoincareDisk. You have to use the sketching tools under the
double headed arrow down the vertical left menu to construct lines and measure angles and such
– you must not use the tools on the upper toolbar (those are Euclidean tools). [You might have
to start with My Computer/local disk/Program files/Sketchpad, etc. – it depends on how the tech
loaded Sketchpad in the first place – it IS worth finding, though.]
43
Enrichment 2:
Parallel Lines in Hyperbolic Geometry
F
Disk Controls
B
G
A
D
E
H
̅̅̅̅ is parallel to every other line showing in the disc.
H𝐴𝐵
̅̅̅̅ intersects H𝐷𝐹
̅̅̅̅ on the circle, these two have a type of parallelism called
Since H𝐴𝐵
“asymptotically parallel”.
̅̅̅̅ .
̅̅̅̅ and H𝐷𝐸
̅̅̅̅ are “divergently parallel” to H𝐴𝐵
H𝐷𝐻
̅̅̅̅ and a point not on it: Point D and we have 3 lines parallel to H𝐴𝐵
̅̅̅̅ through D
So we have H𝐴𝐵
right there on the sketch. This illustrates our choice of parallel axiom. And we now have two
types of parallelism: asymptotic and divergent.
44
Do Workbook Problem 1 right now
Let’s check some axioms, theorems, and definitions just to make sure they’re still true. You
know they’re true, but seeing it is helpful.
45
Enrichment 3:
Vertical angles are congruent.
F
Disk Controls
mGDI = 57.5
B
G
I
A
D
J
H
E
mEDJ = 57.5
I put points I and J on with the Euclidean tool bar “points on arc” at the top of the page AND I
measured these angles using the Hyperbolic Angle Measure from the tool bar on the left under
the double-headed arrow.
46
Enrichment 4:
Triangles and Exterior Angles
F
Disk Controls
L
mBAH = 35.3
B
mAHB = 40.5
mABH = 35.6
A
M
mLBM = 144.4
m1+m2+m3 = 111.44
m1+m2 = 75.80
H
Yes, we have triangles. No, the sum of the interior angles is not equal to 180; it is LESS THAN
180 as promised. The difference between 180 and the sum of the interior angles of a given
Hyperbolic triangle is called the DEFECT of the triangle. In Spherical geometry the difference
between the sum of the interior angles of a spherical triangle and 180 is called the EXCESS of
the triangle.
And, as promised in the Exterior Angle Inequality Theorem, the exterior angle (LBM in our
example above) has a greater measure than either remote interior angle. Thus this theorem is
true in both Euclidean and Hyperbolic geometry.
In fact, its measure is greater than their sum (very non-Euclidean here – remember ALL the facts
after the Parallel Postulate in the Axioms section are Euclidean facts and NOT applicable here in
Hyperbolic Geometry). The Euclidean Exterior Angle Equality Theorem (the exterior angle
measures the sum of the 2 remote interiors) is a Euclidean theorem not a Hyperbolic Theorem.
47
Note that the defect of the triangle is 58.6. (The sum of the angles is 121.4) We use the lower
case Greek letter delta for defect (  ).
Do Workbook Problem 2 right now.
Enrichment 5:
Right triangles
Disk Controls
mCDB = 90.0
C
mDCB = 17.6
mCBD = 18.1
D
B
DC = 1.81
DB = 1.79
A
CB = 2.96
Distance2+Distance2 = 6.48
̅̅̅̅ and the
Here’s a right triangle – the measures of the angles are shown. I built it using an H𝐴𝐵
Hperpendicular bisector of that line (again: Hyperbolic tools on the left menu).
 = 54.3
Does the Pythagorean Theorem hold in Hyperbolic Geometry?
Demonstrably not – see the calculations in the sketch.
48
Enrichment 6:
Distance
The distance (metric) axioms are in the Axioms section
Poincaré Disk Model
Disk Controls
A
M
B
N
Measure the following distance in Euclidean Geometry:
AM
AN
BM
BN
49
You use these Euclidean measurements to calculate the Hyperbolic distance with a formula:
𝐴𝑀⋅𝐵𝑁
the Hdistance from A to B is |ln (𝐴𝑁⋅𝐵𝑀)|
𝐴𝑀⋅𝐵𝑁
𝐴𝑁⋅𝐵𝑀
is called the “cross product”.
The absolute value of the natural log of the “cross product” is a very clever way to measure
distances.
Let’s look at some consequences of this formula.
Poincaré Disk Model
Disk Controls
Euclidean distances
MA = 0.86 in.
BM = 2.07 in.
AN = 1.91 in.
BN = 0.69 in.
Hyperbolic distances
MA = 35.48
HDAB
MB = 37.37

ln
MABN
BMAN

= 1.89
NA = 36.38
NB = 33.50
EG AB = 1.25 in.
A
AB = 1.89
M
B
N
Here’s a picture with Hyperbolic distances on the left and Euclidean distances on the right. The
hyperbolic distances were measured using the hyperbolic distance tool on the left and the
Euclidean distances were measured using the Euclidean distance tool on the top menu. On the
right, the calculation for Hyperbolic distance is shown. Remember that the calculation uses
Euclidean distances. The Hyperbolic tools do the calculation automatically for you.
50
In Euclidean Geometry the distance between points on a segment is fixed and independent of
location in the plane. In Hyperbolic Geometry, however, you have an interesting stretching of
calculated distances that depends on whether the points are close to the center of the disc or close
to the edge of the disc. Points can be the SAME Euclidean distance apart and have different
Hyperbolic distances depending on their location in the disc. This is a function of the distance
formula.
Here’s an illustration with two points that are .09 apart in Euclidean geometry and located in two
different spots in the disc. Note that the Hyperbolic distances are different and the points are
further apart out near the edge of the disc.
Poincaré Disk Model
Disk Controls
A
Euclidean Distance
AB = 0.09 in.
Hyperbolic distance
AB = 0.53
B
M
Euclidean distance
N
.09
Hyperbolic distance .52
51
Poincaré Disk Model
Disk Controls
A
B
N
Euclidean Distance
AB = 0.09 in.
Hyperbolic distance
AB = 0.12
M
Euclidean distance
.09
Hyperbolic distance
.12
Notice that the Hyperbolic distance depends on WHERE you are in the disc. Points that are the
same Euclidean distance apart in different locations on the disc are different Hyperbolic
distances apart. Problems 3 and 4 in the Workbook Homework explore this fact.
52
Enrichment 7:
Quadrilaterals
Poincaré Disk Model
Disk Controls
mIGH = 51.1
mJHG = 51.0
GH = 2.59
G
H
JH = 0.93
GI = 0.93
mGIJ = 90.0
mHJI = 90.0
J
I
IJ = 1.97
This is the only kind of quadrilateral that you can have in all 3 of the big geometries:
a Saccheri Quadrilateral.
Review the definition and the properties.
Do Problem 5 in the Workbook now.
53
Enrichment 8:
Each of the following is logically equivalent tot the Euclidean Parallel Postulate. Thus each of
the following are NOT true in Hyperbolic Geometry. I’ve illustrated below how one equivalence
is not true in HG. The others are Workbook Problem 5.







The area of a triangle can be made arbitrarily large.
The angle sum of all triangles is a constant.
The angle sum of any triangle is 180.
Rectangles exist.
A circle can be passed through any 3 noncollinear points.
Given an interior point of a circle, a line can be drawn through that point intersecting
both sides of the angle.
The perpendicular distance from one of two parallel lines to the other is always bounded.
Illustration:

Two parallel lines are everywhere equidistant.
Poincaré Disk Model
This sketch depic ts the hyperbolic plane H2 us in g the Poincaré disk model. In this model, a line through
tw o poin ts is def ined as the Euc lidean arc pas sing through the points and perpendic ular to the c irc le .
Us e this document's custom tools to perform c onstructions on the hyperbolic plane, comparing y our findings
to equivale nt constructions on the Euc lidean plane.
Dis k Controls
Lines LK and IJ are parallel. I built perpendiculars
to LK and points at the intersection of the
perpendiculars on IJ. In measuring the distance -the "perpendicular" distance -- note that the lines
are NOT equidistant. Equidistance parallel lines
are a Euclidean phenominon.
I
D
J
L
F
LI = 0.76
G
K
E
KJ = 1.27
54