UH
Non-Euclidean Geometries
Introduction
A Flexible Geometry
Some Finite Geometries
Three Point Geometry
Five Point Geometry
An Incidence Geometry
Taxicab Geometry
Spherical Geometry
Hyperbolic Geometry
Answer to exercises
Introduction
An axiomatic system is a formalized foundation for a subject. Axiomatic systems for geometry have the following structure:
Underlying foundations:
The language of logic,
The rules of proof,
The axiomatic system
Undefined terms,
Axioms,
Definition, and
Theorems.
Understanding a system requires modeling, exploration, concentration, and arguments. We’ll start with simple systems and we’ll build up to large complicated geometrical systems.
Undefined terms are a small set of nouns or adverbs that start the whole program off.
We take these as basics and build with them.
Axioms are true statements that lay down the rules of association for the undefined terms.
It is almost always possible to visualize the undefined terms and to build a model of the geometry using the axioms.
Definitions build on the axioms and the undefined terms. They bring richness to the system.
Theorems are truths about the system that require a proof.
We’ve seen one of the bigger geometries: Euclidean Geometry. All the other geometries are in a big set called “Non-Euclidean Geometries”.
We’ll look at some petite geometries. Some are so small, they’ll fit on one page!
And we’ll look at Taxicab Geometry – that’s a famous one. Then we’ll move into Spherical Geometry and finish with Hyperbolic Geometry. These are two big geometries and are as important today as Euclidean Geometry.
A Flexible Geometry **
Undefined terms: point, line, on
Axioms: A1 Every point is on exactly two distinct lines.
A2 Every line is on exactly three distinct points.
Models:
There are lots of very different models for this geometry.
Here are two:
One has a finite number of points and the other has an infinite number of points.
Model 1: 3 points, 2 lines
Points are dots and lines are S-curves. One line
is dotted so you can tell it from the second line. Nobody ever said “lines” have to be straight things, you know.
Note, too, that there are only 3 points so my lines are composed of some material that is NOT points. Some non-point stuff. Luckily they’re undefined terms so I don’t have to go into it.
Model 2: an infinite number of points and lines
This is an infinite lattice. Each line is slanted at 45° and has 3 points along it. It continues forever left and right
Your Model:
You will be asked to invent a model of this on your own in your homework
Ideas for
Definitions:
Biangle – each two-sided, double angled half of the first model…like a triangle but only two points. Do biangles exist in Euclidean geometry (ah, no…check the axioms…two lines meet in exactly ONE point in Euclidean geometry.)
Quadrangle – each diamond-shaped piece of the second model
Parallel lines – Parallel lines share no points.
The second model has them; the first doesn’t. How many lines are parallel to a given line through a given point NOT on that line in the second model? (two! This, too, is really different than Euclidean Geometry).
Collinear points – points that are on the same line.
Midpoints – are these different from endpoints in a way that you can explicate in a sentence for Model 2? Does it make sense to have a “distance” function in this geometry – maybe not…maybe this is something we’ll just leave alone.
What do you notice that cries out for a definition in your model? Making a suggested definition is a homework question.
Theorems: Consider the following questions and formulate some proposed theorems (called “conjectures” until they’re proved)
Flexible Geometry Exercise:
Are there a minimum number of points?
Is there a relationship between the number of points and the number of lines?
Why is this a Non-Euclidean Geometry?
**This geometry is introduced in Example 1, page 30 of
The Geometric Viewpoint: a Survey of Geometries by Thomas Q. Sibley;
1998; Addison-Wesley (ISBN 0-201-87450-4)
Some Finite Geometries
Next we have 2 geometries that are much more closely specified by their axioms than
“A Flexible Geometry”. In fact, there are a specific number of points in each one.
The following two examples of finite geometries each has a model with a different number of points, 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.
The Three Point Geometry
Undefined terms: point, line, on
Axioms: A1 There are exactly three distinct points.
A2 Two distinct points are on exactly one line.
A3 Not all the points are on the same line.
A4 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.
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.
The proof for Theorem 2 is a homework problem.
The Five Point Geometry
Undefined terms point, line, on
Axioms A1 There are exactly five points.
A2 Any two distinct points have exactly one line on both of them.
A3 Each line is on exactly two points.
Models
Points: {P1, P2, P3, P4, P5}
Lines: {P1P2, P1P3, P1P4, P1P5, P2P3, P2P4, P2P5,
P3P4, P3P5, P4P5}
Note that the lines crossover one another in the interior of the “polygon” but DO NOT intersect at points. There are only 5 points!
Possible Definitions
Triangle -- a closed figure formed by 3 lines. An example: P2P1P4 is a triangle.
How many triangles are there?
Quadrilateral – a closed figure formed by 4 lines. An example: P2P5P4P3 is a quadrilateral. How many quadrilaterals are there?
Distance – probably not meaningful since every point is connected to every other point.
Angle measurement – probably not meaningful – it’s another distance and this geometry is too small for distance to mean much.
Parallel lines – two lines are parallel if they share no points.
Note that line P1P2 is parallel to line P4P5. So are P3P4 and P2P5…
5 Point Geometry Exercise:
How many pairs of parallel lines are there?
Other possible defnitions:
Collinear points
Interior
Plane
Theorem
Each point is on exactly 4 lines.
An Incidence Geometry
Undefined terms: point, line, on
Axioms: A1 There is exactly one line on any two distinct points.
A2 Each line has at least two distinct points on it.
A3 There are at least three points.
A4 Not all the points lie on the same line.
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
Angle measurement
Intersecting lines
Triangles
Quadrilaterals
Between or interior
Concurrent lines
Theorems:
Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.
Theorem 2: Each point is on at least two lines.
Theorem 3: There is a triple of lines that do not share a common point.
An Incidence Geometry, continued
A six point model:
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.
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:
AB BC CD DE EF
AC BD CE DF
AD BE CF
AE BF
AF
A1 There is exactly one line on any two distinct points. See the list
A2 Each line has at least two distinct points on it. See the “endpoints”.
A3 There are at least three points. There are 6 which is “at least 3”.
A4 Not all the points lie on the same line. See the list.
Theorem 1: If two distinct lines intersect, then the intersection is exactly one point.
Theorem 2: Each point is on at least two lines.
Theorem 3: 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.
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.
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!
An Incidence Geometry, continued
A Model with an infinite number of points and lines:
Points will be {(x, y) ( x2 + y2 < 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.
H is a point in the circle’s interior and IS a point in the geometry.
We cannot list the number of lines – there are an infinite number of them.
Checking the axioms:
A1 There is exactly one line on any two distinct points.
This model is a subset of Euclidean geometry and the axiom holds.
A2 Each line has at least two distinct points on it.
Each line has an infinite number of points by Euclidean Axioms.
A3 There are at least three points.
The unit disc has an inifinite number of points.
A4 Not all the points lie on the same line.
True
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.
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: If two distinct lines intersect, then the intersection is exactly one point.
Inherited from Euclidean Geometry.
Theorem 2: Each point is on at least two lines.
Each point is on an infinite number of lines.
Theorem 3: There is a triple of lines that do not share a common point.
FE, GC, and AD for example.
Incidence Geometry Exercise:
Find at least one way that this geometry is like Euclidean Geometry.
Find three ways that this geometry is different from Euclidean Geometry.
Taxicab Geometry
another non-Euclidean geometry
Axiomatic Structure:
undefined terms: point, line, plane
axioms:
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. Every plane contains at least three non-collinear points.
B. Space contains at least four non-coplanar points.
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(.
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
next in Euclidean Geometry would be SAS, Axiom 15, but we’ll discover that we don’t have this axiom in Taxicab Geometry…it’s NOT true.
Taxicab distance, TCG coordinates, and angle measurement:
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.
We will be working in the Cartesian Plane
so our points will come with algebraic coordinates (a, b).
The distance between two points P1 and P2 with coordinates similarly subscripted is:
TCGD* = [pic]
*TaxiCabGeometry distance. NOTE this is a different formula from Euclidean Distance!
Does order of subtraction really matter? Could I have written: [pic] YES!
Example:
What is the Euclidean distance from (3, 2) to (−1, 9)?
The formula: [pic] which rounds to 8.1
What is the TCGD between the same points?
[pic]
These are very different distances between the same two points.
TCG Distance Exercise:
Let’s begin looking at some distances. Put the points on the graph before you calculate the distances:
| | | |
|Points |Euclidean Distance |Taxicab Distance |
| | | |
|(0, 0) to (4, 0) | | |
| | | |
|(0, 0) to (1, 3) | | |
| | | |
|(0, 0) to (2, 2) | | |
| | | |
|(0, 0) to (0, 4) | | |
| | | |
|(3,3) to (6, 7) | | |
| | | |
|(3, 3) to (6, 3) | | |
| | | |
|(3, 3) to (3, 6) | | |
| | | |
|(5, 0) to (8, 0) | | |
| | | |
|(5, 0) to (7, 8) | | |
| | | |
|(5, 0) to (5, 4) | | |
[pic]
Do you see an easy physical description of how to calculate the TCG distance?
Do you have a conjecture about when the EG distance and the TCG distance is the same?
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.
Taxicab Geometry Coordinate Exercise:
We will use the line y = x. Sketch it in here and put on points with x values: (8, (3, 0, 2, 5.
[pic]
Fill in the following table that shows the Cartesian coordinates, Euclidean Geometric coordinate, and the TCG coordinates for the distances between the following points:
Recall: the Euclidean geometric coordinate formula is [pic]
Our goal with this exercise is to find the formula for the TCG coordinate.
One hint: it’s the x value times something…let’s find “something”
| point in | | |
|Cartesian coord |EG coord |TCG coord |
| | | |
|(0, 0) |0 |0 |
| | | |
|(2, 2) |2[pic] |4 |
| | | |
|(5, 5) |5[pic] |10 |
| | | |
|((3, (3) |(3[pic] |(6 |
| | | |
|((8, (8) |(8[pic] |(16 |
What’s your conjecture about the coordinate formula?
Try the same exercise with the line y = (2x +1 Sketch in the line: show the y-intercept and the points with x values: −4, −2, 0, 1, 3, and 4.
[pic]
| | | |
| |EG coord – on your own |TCG coord |
|y intercept | | |
| | |0 |
|(0, 1) | | |
| | | |
|(4, (7) | |12 |
| | | |
|((2, 5) | |−6 |
| | | |
|(1, (1) | |3 |
| | | |
|((4, 9) | |−12 |
| | | |
|( 3, (5) | |9 |
What’s your conjecture?
A11. The Angle Measurement Postulate:
To every angle there corresponds a real number between 0( and 180(.
Angles will be measured with protractors just as though you’re measuring a Euclidean angle.
[pic]
Angle exercise 1:
Put a 3 – 4 – 5 triangle on the graph. Make the triangle be off the axes out in Q1.
What are the measures of the angles? You ought to know these by heart now.
Does it seem like trigonometry is safe and sound and will work with this system?
Hint measure the side lengths in TCGD and talk about the sine being the length of the side opposite divided by the length of the hypotenuse.
Did I lead you astray talking about the triangle in Euclidean terms – I bet so. We have to be careful of the context here. I’ll be marking every question on the final so you know WHICH geometry the question is about.
[pic]
Angle exercise 2:
What about an isosceles right triangle with the base parallel to the x axis?
Do our trig values hold there?
Use the points (7, 7), (2, 3) and (12, 3). What are the 3 angles measures?
What are the Euclidean side lengths?
What are the TCGD side lengths?
Does traditional Trigonometry work? Can you use inverse trig functions freely in Taxicab Geometry.
In actuality there’s a TCG trigonometry that we’ll look at as soon as we get some geometric shapes established. It’s mostly a diversion and not really useful, but it’s fun to contemplate and it won’t be on the final, promise.
Geometric Shapes:
Lines: look like lines, no problem there. However, the interaction of points on a line with other points and with other lines is somewhat changed in TCG.
Betweeness:
We have a concept in EG called “between” – B is between A and C iff AB + BC = AC.
In Euclidean Geometry, if you have points A and C on a line, the points “between” them are also on the line. They are collinear with the segment endpoints.
Now, since we’re using a different formula for distance, you should not be startled if different points add up properly to be “between” in a sense particular to TCG.
Betweeness exercise: Let’s look at TCG betweeness
[pic]
Point A is (1, 7) and point C is (11, 1). How far is point A from point C?
Well, now, I have to clarify this question --
Which is to say, what is AC in Taxicab Geometry? in Euclidean Geometry?
[pic]
Mark the midpoint of the line…(6, 4)
How far is the midpoint from each endpoint in TCG?
So the midpoint of the line segment is between A and C. 8 + 8 = 16. It “adds up” correctly.
I’ll get you started: using point (11, 7) as C
AC = 10 and CB = 6 so AC + CB = 16 for this C. And (11, 7) is between A and C in the TCG sense of between. So is: (6, 6)…do you see how?
Find at least 5 additional points that add up correctly according to the formula. The third point will be called C and AC + CB = 16. (hint: work on both sides of the line segment…)
Extend this to cover ALL the points that are “between” A and C in a TCG sense.
TCG Perpendicular Bisector Exercise:
Using the same drawing let’s put in the perpendicular bisector to segment AC…this is the set of all points equidistant from the endpoints. In Euclidean Geometry, the perpendicular bisector is a straight line. In TCG some of them have bends and are NOT straight. All the points are equidistant from the endpoints though.
[pic]
Sketch in points that are equidistant from A and C.
Hint: inside the “betweenness rectangle” is slightly different from outside as far as the distances and the look of the points go.
Triangles:
Since we measure angles the same way, the sum of the interior angles is still 180(.
But let’s check on a few other items:
Equilateral triangles exercise:
Sketch an EG equilateral triangle with side measure 4 and the left vertex at the origin.
What are the coordinates of the apex angle?
Is it TCG equilateral? nope…the sides are not length 4, they are an irrational length.
Sketch a TCG equilateral triangle side length 4 with the base starting on the left at
(5, 0) on top of the x axis…do the angles measure 60(? What do the angles measure?
What are the coordinates of the apex angle?
[pic]
Then sketch the one with vertices: (1, 7) (4, 8) and (2, 10). It’s TCG equilateral BUT…. EG isosceles.
What can you conclude about TCG equilateral triangles? How are they like the familiar Euclidean ones? How are they different?
Now let’s sketch in a triangle with a horizontal length of 4 and a vertical length of 4 and one vertex over in the upper right of the graph paper…what is the hypotenuse length?
In TCG: 8 in EG: [pic]
Is this triangle congruent to the TCG equilateral triangle of side length 4 that is to the right on the x axis? Why or why not?
Can you say the triangles have the quality SAS?
Is SAS an axiom in this geometry?
Do you see why this geometry is non-euclidean?
We’ll have to use the old fashioned definition of congruence – 3 corresponding congruent sides and 3 corresponding congruent angles for congruence in Taxicab Geometry.
SAS will NOT do as a shortcut to declaring congruence at all. Note that it is NOT on the axiom list – I stopped short of this axiom. Taxicab Geometry and Euclidean geometry have only the axioms up to SAS in common.
Circles:
A circle is the set of all points that are equidistant from a given point called the center of the circle.
Happily, we do have circles in TCG. Let’s figure out what they look like!
Sketch the TCG circle centered at the origin with radius 4:
(Check back on page 3 for several points that measure 4 from the origin before you get started!)
[pic]
How many times have you told someone that you’ll pick them up downtown right in front of their building and that you’ll “circle” the block til you see them? Does this language reflect Euclidean Geometry or Taxicab Geometry?
Module 5
Spherical Geometry
Suppose we change the surface upon which we are working?
Instead of a plane, we’ll work on the surface of a sphere.
Undefined terms: point, line, on
Let “lines” be represented by “great circles”. In navigation and geography, then, lines are meridians of longitude. It will be a definition-level requirement that the lines in the geometry are “great circles”
Before we go further, we will use a model to explore some of the changes working on a sphere causes.
A Model for Spherical Geometry
A physical model can be made using a smooth sphere and lines that will stay in place (rubber bands are good for this):
Find a point on the model (on a semantic note: would you have gotten confused if the question were “find a point in the model”?) Is the center of the sphere a point in/on the model?
Find two points that are a maximum distance apart.
Find a pair of perpendicular lines. How many points of intersection are there?
Find a pair of parallel lines.
Find a triangle. Find a small triangle and find a really big one.
Find a quadrilateral.
Find a circle.
Decide if you’re in Kansas anymore.
An algebraic model is the surface of a sphere in three dimensions. The usual formula for the shape:
{ (x, y, z) ( (x – h)2 + (y – k )2 + (z – j)2 = r2 }
where, “r” is the radius of the sphere and ( h, k, j ) is the location of the center.
The surface area of a sphere is 4 (R2 and while we will be working on the surface only, it is necessary to have a notion of where the center is in order to measure the radius. Please spend time imagining the great circle construction using the XY plane and the sphere centered at (0, 0, 0) with radius length one. The sphere itself is often called The Unit Sphere and its intersection with the XY plane results in The Unit Circle – these are common nicknames for these two sets of points.
Axioms for Spherical Geometry
First return to Euclid’s five postulates. Check each one carefully and make modifications as needed.
Next, check the SMSG axioms. There will have to be modifications here, too. In light of what is or is not present, which theorems from the two Euclidean geometry sections might be invalid or in need of modification?
Definitions**
Antipodal points – Antipodal points are the points of intersection between a sphere and a line that includes the center of a sphere.
Biangle or Lune – A lune is one of the regions between two great circles and the two points of intersection of these great circles which are the vertices of the lune. (note that these vertices will be antipodal points.
Distance – see below. It has it’s own section!
Geodesic – The geodesic is the curve that minimizes the distance between two points.
Great circle – A great circle is the intersection of a sphere and a plane that includes the center of the sphere.
Line – a “great circle” on the surface of a sphere.
Lune – A lune is a polygon on the sphere formed by the intersection of any two lines.
Pole – A pole of a great circle is one of the endpoints of a diameter of the sphere that is perpendicular to the plane of the great circle. Poles come in pairs.
Quadrilateral – A four-sided figure.
Radian measure – a unit-free angular measure in which 180( = Π.
Saccheri quadrilateral – A Saccheri quadrilateral is a biperpendicular quadrilateral with equal side lengths (legs).
Triangle – In this section, we will focus on “small” triangles: Given three points no pair of which are antipodal for vertices, a small triangle has side lengths that are the shorter segments of the great circles that join the vertices.
** these are all in the glossary, too, though some have been expanded upon here.
Distance Formula and Radian measure
The distance between two points, A and B, is the product of the sphere’s radius and the measure of the central angle subtended by A and B, denoted d(A,B) = Rα ( with A and B are on the sphere’s surface, C is the center of the sphere, α is the m( ACB in radian measure, and R is the radius of the sphere). Essentially, this says that if A and B are points on the sphere, then the distance between them is the arc length along the great circle connecting them (conventionally, the shortest distance which is called the geodesic).
The procedure for measuring an angle is to project the lines up to a plane that is tangent to the sphere at the vertex of angle and measure in the plane and in radian measure. (Angular measurement can be in degrees, but for ease and by common agreement, upper division levels of mathematics use radian measure. )
Attached are some worksheets for practice in two dimensions at measuring angles in radian measure.
Radian measure worksheet
Radian measure worksheet
Radian measure worksheet
Lunes or Biangles**
This is the simplest polygon on our surface. Build a lune on your model.
Keep in mind that the vertices are antipodal points. Can you observe anything about the vertices?
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:
Area of a lune = 2R2(
Where alpha is the lunar angle measure.
We’ll work out a way to see this without actually proving it:
Decomposition Lunar Area of
Number of lines into congruent lunes angle measure the shape
None none 2Π 4ΠR2
One two ½ of 2Π ’ (
Two four
Three
Four
.
.
Eight
.
.
n
** I’ll probably use these terms interchangeably; some authors favor one over the other…it’s too new to have gotten fossilized.
Triangles
In Spherical geometry, the angle sum of a triangle exceeds π ( 180() and the excess is proportional to the area of the triangle. Do some work with a REALLY, REALLY big sphere and you’ll see this. This fact is too startling to leave unproven.
Giraud’s Theorem
The area of a spherical triangle is R2 (r + g + b – π ).
Proof
Let T be any triangle with angles measuring A, B, C. Consider the lines that form each vertex. . At each vertex there are two lunes formed; one is the lune that contains T and the other will be called the antipodal lune that contains an antipodal copy of T, called T*. These triangles are congruent.
There are three pairs of such lunes; each decomposed into two triangles. These pairs of lunes overlap one another rather than decomposing the sphere into disjoint lunes.
Since the area of a lune is 2R2(, we may add the areas of these lunes to arrive at the surface area of the sphere. As we add them, of course, we are adding the areas of four extra copies of the triangle T in the following fashion:
Area Lune 1 includes T 2R2A +
Area Lune 2 includes T* 2R2A +
Area Lune 3 includes T 2R2B +
Area Lune 4 includes T* 2R2B +
Area Lune 5 includes T 2R2C +
Area Lune 6 includes T* 2R2B
4(R2 + 4 Area T = 4R2( A + B + C)
Solving the equation for Area T, we find that
Area T = R2 ( A + B + C ( ( ) (
Implications of this theorem are:
( In Spherical geometry, triangle area is based on angle measure rather than side length.
( The sum of the interior angles of a triangle is more than 180( (ie, () !
To see this, backsolve the conclusion of Giraud’s Theorem for the sum of the interior angles:
Area T = R2 ( A + B + C ( ( )
A + B + C = ( + R-2 Area T.
( Similarity is a meaningless concept in Spherical geometry.
( The Exterior Angle Theorem isn’t true here.
Quadrilaterals
Summit angles of a Saccheri quadrilateral are congruent and obtuse.
The Quadrilateral Associated with Given Triangle
What’s New and What’s Not True
No parallel lines
Triangle interior measures is different, area formulas are different
AAA becomes a congruence condition; similarity is meaningless
AAS is not a congruence condition because many triangles can have two right angles
Betweeness of points on a line is gone
Intersection of two lines is two points. There is a new shape: biangles
Hyperbolic Geometry
Undefined terms: Point, line, on
Axioms: Initially, let’s modify Euclid’s structure; next, we’ll look hard at the SMSG structure.
Definitions: Parallel lines
Asymptotically parallel
Divergently parallel
Angle measure
Distance
Saccheri Quadrilateral
Quadrilateral Associated with A Triangle
Theorems Sum of the Interior Angles Theorem
Exterior Angle Theorem
Saccheri Quadrilateral Theorems
Triangle Congruence Theorems
Conventional Surfaces and Models**
We will use the Poincaré Disk:
Points will be represented by the set of all points on the open disk, D. The algebraic definition of open disk is { (x, y ) | x2 + y2 < r2}. Euclidean arcs that intersect the excluded boundary orthogonally will represent lines. By convention, any diameter will be considered a circle of infinite radius and will be considered a line.
Definitions
Distance Formula
Let X and Y be points in the disk. P and Q will be the points where the line through X and Y intersect the disk. The cross ratio of X and Y is
The distance from X to Y is the logarithm of the absolute value of the cross ratio of X and Y.
d(X,Y) = log | C (X,Y) |
Two line segments, AB and CD, are congruent if d(A,B) = d(C,D).
Parallel lines divergently and asympototically
** See the website, , for very lovely pictures and explanations of alternate models including a three dimensional model.
Shapes and their Properties
Triangles
Quadrilaterals
Circles
Parallel and Intersecting lines
Theorems
Sum of the interior angles of a triangle
Saccheri Quadrilaterals
What is the same and what is no longer true?
Answers to exercises
Three Point Geometry exercise:
[pic]
6 Point Geometry Exercise:
Note that line P1P2 is parallel to line P4P5.
How many pairs of parallel lines are there?
There are 30 pairs of parallel lines. Each line is parallel to three others.
Here are some examples:
P1P2 is parallel to P3P4, P3P5, and P4P5.
P1P5 is parallel to P2P3, P2P4, and P3P4.
P2P5 is parallel to P1P3, P1P4, and P3P4.
Flexible Geometry Exercise:
Are there a minimum number of points?
There are 3 points to the “minimum” model.
Is there a relationship between the number of points and the number of lines?
The ratio of points to lines is 3:2.
Why is this a Non-Euclidean Geometry?
The axioms are totally different from Euclidean Geometry.
Note that this geometry has several distinctly different models – and Euclidean Geometry has essentially only one model. (This is a new fact for you!)
Incidence Geometry Exercise:
Find at least one way that this geometry is like Euclidean Geometry.
The axiomatic structure, the framework, is the same format:
undefined terms
axioms
definitions
theorems
The points and lines are actually from Euclidean Geometry and are actually called points and line segments.
Find three ways that this geometry is different from Euclidean Geometry.
The axioms themselves are different.
There are at least two different models; Euclidean Geometry has only one model.
There are two types of parallel lines with infinite point model – this is totally
non-euclidean. The finite model has two lines parallel to a given line through a point not on the line; this, too, is different.
TCG Distance Exercise:
Let’s begin looking at some distances. Put them on the graph before you calculate the distances
| | | |
|Points |Euclidean Distance |Taxicab Distance |
| | | |
|(0, 0) to (4, 0) |4 |4 |
| | | |
|(0, 0) to (1, 3) |[pic] |4 |
| | | |
|(0, 0) to (2, 2) |[pic] |4 |
| | | |
|(0, 0) to (0, 4) |4 |4 |
| | | |
|(3,3) to (6, 7) |5 |7 |
| | | |
|(3, 3) to (6, 3) |3[pic] |3 |
| | | |
|(3, 3) to (3, 6) |3[pic] |3 |
| | | |
|(5, 0) to (8, 0) |3 |3 |
| | | |
|(5, 0) to (7, 8) |[pic] |10 |
| | | |
|(5, 0) to (5, 4) |4 |4 |
I’ve graphed the first 7 points. You graph the rest!
[pic]
Do you see an easy physical description of how to calculate the TCG distance?
go horizontally the difference in the x’s and vertically the difference in the y’s and add the number of “steps”.
Do you have a conjecture about when the EG distance and the TCG distance is the same?
They’re the same on purely vertical (slope undefined) or purely horizontal (slope zero) and different on lines with slope between 0 and undefined.
Note that the first five points are equidistant from the origin in TCG – all are distance 4 from the origin.
Taxicab Geometry Coordinate Exercise:
The formula is x([pic]).
So on the line y = 3x + 12 the coordinate for (2, 18) is 8.
Angle exercise 1:
Put a 3 – 4 – 5 triangle on the graph. Make the triangle be off the axes out in Q1.
What are the measures of the angles? You ought to know these by heart now.
Does it seem like trigonometry is safe and sound and will work with this system?
Hint measure the side lengths in TCGD and talk about the sine being the length of the side opposite divided by the length of the hypotenuse.
The actual and correct angle measure is 36.87° (approximately). But if we casually apply trigonometry, we get another different and incorrect value. You have to use protractors or go between EGC and TCGD to get angle measurements versus linear distances.
Angle exercise 2:
Use the points (7, 7), (2, 3) and (12, 3). What are the 3 angles measures? 2 45’s and a 90°
What are the Euclidean side lengths? 10 and [pic] twice
What are the TCGD side lengths? 10 and two 9’s.
Does traditional Trigonometry work? Can you use inverse trig functions freely in Taxicab Geometry? No you won’t get 45° if you use arcsin and the TCGD measures.
Betweeness exercise:
The TCG distance is: 16
The Euclidean distance is [pic]
The points between A and C are a rectangle – millions and billions of points – the rectangle is bounded by (1, 1) and (1, 7) and (11, 7) and (11, 1). It’s the green box below – except that you have to imagine that the box lines up exactly with the grid.
Points line (2.5, 3.9) are “between” because the math for the formula works out right even though it’s not collinear with the line AC.
[pic]
This a situation that is called being “metrically between” as opposed to Euclidean between. Euclidean between is just the open line segment from A to C.
TCG Perpendicular Bisector Exercise:
Using the same drawing let’s put in the perpendicular bisector to segment AC…this is the set of all points equidistant from the endpoints. In Euclidean Geometry, the perpendicular bisector is a straight line. In TCG some of them have bends and are NOT straight. All the points are equidistant from the endpoints though.
[pic]
Connect the inserted dots to see the crooked perpendicular bisector, which in TCG is not a line but a collection of points equidistant from A and C. Note that it DOES look like the line you’re expecting INSIDE the betweenness box and it changes once it gets to points that are no longer between A and C.
Equilateral triangles exercise:
The EG equilateral triangle has vertices: (0, 0) (2, 2[pic] ) and (4, 0)
The TCG equilateral triangle has vertices: (5, 0) (7, 2) and (9, 0)
The apex angle in the EG one is 60(; in the TCG on it is 90( and the base angles are 45(.
Clearly some TCG equilateral triangles are not equiangular. Some TCG equilaterals are only isosceles in EG.
The isosceles triangle that is TCG 4, 4, 8 in the upper right has interior angles 45(, 45(, 90(. Thus it can be said that both – non-congruent triangles have 4 - 90( - 4…SAS. In other words…SAS doesn’t work as a way to declare triangles congruent in TCG.
This is a huge difference between EG and TCG.
-----------------------
C
A
[pic]
C
A
C
A
C
A
C
A
Between: TCG
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