Spherical Geometry – Part II
Construction on the Sphere
Step 1
Draw two different points on your sphere. Label them A and B.
Step 2
Pick either of the two ruled edges of your spherical ruler and align it with your two points.
Step 3
Draw the line that connects A and B along the spherical ruler and extend it as far as possible
in both directions.
Investigate:
1.
You have just created a great circle on a sphere. How is the radius of the great circle related to
the radius of the sphere?
Great circles are the “lines” on a sphere.
2.
A. Into how many arcs (sections) do points A and B divide your great circle? ___________
B. How many of these arcs are finite? ______________
C. How many are infinitely long? ______________
3.
A. How many great circles can you draw through two points on a sphere? __________
B. Is your answer true for any two points on a sphere? (think of the North and South Pole on the
globe. Points such as these are called polar points.) ________________
More Construction
Step 5
Shift the spherical ruler and draw another “line” or great circle.
Investigate
5.
When two great circles intersect, in how many points do they intersect? __________
6.
Can you rotate or shift the spherical ruler and draw a great circle that is parallel to the great circle
that goes through points A and B?
Comparing Planar and Spherical Geometry
Complete the table below to compare and contrast lines in the system of plane Euclidean geometry and
lines (great circles) in spherical geometry.
On the plane
On the sphere
1. Is the length of a line finite or
infinite?
2. Describe the shortest path
that connects two points.
3. Can you extend a line
forever?
4. How many parts (and are
they finite or infinite) will two
points divide a line?
5. How many lines pass through
any two different points?
6. How many lines are parallel
to a given line and pass through
a given point not on the given
line?
7. If three points are collinear,
exactly one is between the other
two. (True or false)
For each property listed from plane Euclidean geometry, write a corresponding statement for spherical
geometry.
8.
Two distinct lines with no point of intersection are parallel.
9.
Two distinct intersecting lines intersect in exactly one point.
10. A pair of perpendicular lines divides the plane into four infinite regions.
11. A pair of perpendicular lines intersects once and creates four right angles.
12. Parallel lines have infinitely many common perpendicular lines.
13. There is only one distance that can be measured between two points.
14. There is exactly one line passing through two points.
OVER
Choose one of the following answers for each question: A)
B)
C)
D)
true on a plane
true on a sphere
true on both a plane and a sphere
true on neither a plane or a sphere
_____15. A line is an infinite set of points.
_____16. A line is continuous (no “holes” or gaps).
_____17. Through any two points, there is exactly one line.
_____18. There exists at least one pair of points through which more than one line can be drawn.
_____19. A polygon may have two sides.
_____20. Each angle of an equilateral triangle must be 60˚.
_____21. Each angle of an equilateral triangle may be 45˚.
_____22. Each angle of an equilateral triangle may be 120˚.
_____23. A line is bounded. (that is, it can fit into a closed box)
_____24. There is no greatest distance between two points.
_____25. Two lines can share no points.
_____26. Two distinct lines can share two points.
_____27. Two distinct lines can share more than two points.
_____28. The sum of the angles of a triangle is always the same number.
_____29. A triangle can have at most one right angle.
_____30. Three lines may be perpendicular to each other (that is, line a  line b  line c  line a)
_____31. Three lines may be parallel to each other.
_____32. Three lines may intersect in three points. (each of the lines intersects the other two lines)
_____33. Three lines may intersect in two points. (each of the lines intersects the other two lines)
_____34. Three lines may intersect in four points.
_____35. Vertical angles are congruent.
Polygons on a Sphere
Construction
Step 1
Draw a great circle where the equator would be on the globe.
Step 2
Draw a great circle going through the North and South Poles.
Step 3
Rotate the spherical ruler to draw another great circle through the North and South Poles.
Investigate
1.
What size angle is formed where the “North-South” great circle intersects the “equator”? _______
2.
When two “lines” (great circles) on a sphere are perpendicular, how many right angles are
formed? ________
3.
Locate a triangle that you have created. Do the measures of the 3 angles add up to 180˚? _______
4.
Is it possible to draw a triangle on a sphere that has 2 right angles? _________
5.
Is it possible to draw a triangle on a sphere that has 3 right angles? _________
6.
On a plane, the fewest number of sides that a polygon can have is three. You have already made a
polygon with three sides on the sphere. Can you make a polygon on the sphere with only two
sides? _________
Euclidean/Non-Euclidean Geometry
About two thousand years ago, Euclid summarized the geometric knowledge of his day. He developed this
geometry based upon ten postulates. The wording of one of his postulates, known as the parallel
postulate, was very awkward and received much attention from mathematicians. These mathematicians
worked diligently to prove that the conclusions in Euclidean geometry were independent of this parallel
postulate. A mathematician named Saccheri wrote a book called Euclid Freed of Every Flaw in 1733. His
attempt to show that the parallel postulate was not needed actually laid the foundation for the
development of the two branches of non-Euclidean geometry. Euclidean geometry assumes that there is
exactly one parallel to a given line through a point not on that line. The branch of non-Euclidean geometry
called spherical or Riemannian assumes that there are no lines parallel to a given line through a point not
on that line. The other branch of non-Euclidean geometry called hyperbolic or Lobachevskian geometry
assumes that there is more than one line parallel to a given line through a point not on that line.
Physical models for these geometries allow us to visualize some of their differences. The model for
Euclidean geometry is the flat plane. The model for hyperbolic geometry is the outside bell of a trumpet.
The model for spherical geometry is the sphere.
I.
We have proved that the sum of the angles of a triangle is 180. On a globe, is it possible to have a
triangle with more than one right angle? _________ Is this a Euclidean triangle? _______ Why
or why not? _______________________________________
The sides of this triangle (on the globe) curve through a third dimension. The surface upon which
the triangle is drawn affects the conclusions about the sum of its angles. Euclidean geometry is true
for measurement over relatively short distances (when the surface of the earth approximates a flat
plane). Remember the physical experiences possible when this geometry was developed. The
geometry of Einstein’s theory of relativity is the geometry of no parallel lines (spherical or
Riemannian). Notice that these non-Euclidean geometries are derived from different postulates.
II. A second type of non-Euclidean geometry results when a single
definition is changed. Euclidean geometry defines distance “as the
crow flies.” In other words, distance is the length of the segment
determined by the two points. However, travel on the surface of the
earth (the real world) rarely follows this ideal straight path.
y
On the grid at the right, locate point A with coordinates (-4, -3) and
point B with coordinates (2, 1). Use the Pythagorean Theorem to find
the Euclidean distance between A and B.
Now consider that the only paths that can be traveled are along grid
lines. This distance is called the “taxi-distance.” What is this “taxidistance” from A to B?




Points on a taxicab grid can only be located at the intersections of horizontal and vertical lines.
One unit will be one grid unit.
Therefore, the numerical coordinates of points in taxicab geometry must always be _____________.
The taxi-distance between 2 points is the smallest number of grid units that an imaginary taxi must
travel to get from one point to another.
x
1. Two points determine a line segment. (a segment is the shortest distance between two points)
(a) Draw a taxi segment from point A to point B. What is the length of this
B
segment? _________

(b) Is this the only taxi segment between the two points? ______
If not, how many different taxi segments can you draw between points A
and B? _______

A
(c) In taxicab geometry, do two points determine a unique segment? ________
2. A circle is the set of points in a plane that are the same distance from a given point in the
plane.
(a) On the grid at the right, draw a taxi-circle with center P and a
radius of 6.
(b) Is this the only taxi-circle that can be drawn with this center and
this radius? ______ If not, how many different taxi-circles can
be drawn? _______
P

(c) Can you draw a Euclidean circle without lifting your pencil?
_______ ; the “taxi-circle”? _____ The “taxi-circle” is an
example of discrete mathematics where the sample space is a set
of individual points (not a continuous set such as a number line).
3. A midpoint, M, of a segment, AB , is a point on the segment such that AM = MB.
Q
(a) Find the midpoint of segment PQ.
(b) Is there more than one midpoint? ______
(c) Find the midpoint of segment PT.
T
(d) What conclusion can you make about the number of midpoints in taxicab
geometry?
P
4. A point is on a segment’s perpendicular bisector if and only if it is the same distance from each
of the segment’s endpoints.
(b) Find all points that satisfy
(c) Find all points that satisfy
(a) Find all points that satisfy
this definition in taxicab
this definition in taxicab
this definition in taxicab
geometry for segment DE.
geometry for segment ST.
geometry for segment KL.
S

D

L

E
T

K
(c) What conclusion can you make about perpendicular bisectors in taxicab geometry?
Taxicab Geometry Questions
Name_________________________
Date______________Period_______
1. Find the taxicab distance between (-1, 2) and
y
(3, 5).
2. Draw the “taxi” circle with center at (1, -2) and
y
radius 5.
x
3. Find the midpoint of the segment with
endpoints at (-6, -5) and (7, 4).
y
x
4. Find the perpendicular bisector of the segment
with endpoint at (-3, -5) and (5, 1). y
x
x
Apartment hunting – Jane and George are looking for an apartment in a city where the streets all
follow the grid lines. Jane works as a waitress at a restaurant at J(-3, -1). George works as a
technician at the local television station at G(3,3). They walk wherever they go.
5. They have decided their apartment should be
6. In a moment of chivalry George decides that
located so that the distance Jane has to walk to
the sum of the distances should still be a
work plus the distance George has to walk to
minimum, but Jane should not have to walk any
work is as small as possible. Where should they
farther than he does. Now where could they look
y
look for an apartment?
for an apartment?
y
x
x
7. Jane agrees that the sum of the distances
should be a minimum, but she is adamant that
they both have exactly the same distance to walk
to work. Now where could they live?
y
8. After a day of fruitless apartment hunting, they
decide to widen their area of search. The only
requirement that they keep is that they both be the
same distance from their jobs. Now where should
y
they look?
x
A)
B)
C)
D)
x
True for Euclidean geometry
True for Taxicab geometry
True for both Euclidean geometry and Taxicab geometry
False for both Euclidean geometry and Taxicab geometry
Choose the best answer from the above list for each.
_________9. Every segment has a midpoint.
________10. The distance between two points is a constant.
________11. A circle is a continuous set of points.
________12. A circle is a finite set of points.
________13. Only integers can be coordinates of a point.
________14. All points on the plane have coordinates. or
Every point on the plane has coordinates.
________15. Exactly one segment can be drawn between any two points.
________16. Points on a perpendicular bisector of a segment are equidistant from the endpoints of the
segment.
________17. Not all segments have perpendicular bisectors.
________18. A perpendicular bisector is an infinite set of points.
________19. A perpendicular bisector is a continuous set of points.
________20. A perpendicular bisector is a discrete set of points.
________21. A circle is a discrete set of points.
________22. Exactly one segment is the shortest distance between two points.
Graph Theory – Polygons as Networks
Network theory is a branch of topology founded by Leonhard Euler about 250 years ago. Euler worked on
two topological problems 100 years before topology had been so named. These two problems founded
network theory. One of these problems, the famous puzzle of the Konigsberg bridges, will be discussed
later.
(the following is from the textbook, Glencoe Geometry by Cindy Boyd (1998), beginning on page 559)
The international computer information system, Internet, was called the information highway in the early
1990s. It is the networking of computers, connections, and information. Internet is composed of nodes that
node
allow for information to be transferred from one computer site to another.
The diagram at the right represents a network. Such a diagram
illustrates a branch of mathematics called graph theory in networking,
the points are called nodes, and the paths connecting the nodes are
called edges. Edges can be straight or curved.
●
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edge
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Straight edges can be used to form closed or open graphs. If an edge of a closed graph intersects exactly
two other edges only at the endpoints, then the graph forms a polygon.
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Closed graphs
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Open graphs
However, not all pairs of nodes are connected by an edge in some networks. A network like this is called
incomplete. Therefore, a complete network has at least one path between each pair of nodes.
V
●
U
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●
Y
W
●
A
●X
incomplete
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D●
●B
●C
L
F
●M
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O
J
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N
●G
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complete
H
You have probably seen puzzles that ask you to trace over a figure without lifting your pencil and without
tracing any lines more than once. In graph theory, if all nodes can be connected and each edge of a network
can be covered exactly once, then the network is said to be traceable.
The degree of a node is the number of edges that are connected to that node. The traceability of a network
is related to the degrees of the nodes in the network. An odd node is a node that is the endpoint of an odd
number of edges. An even node is a node that is the endpoint of an even number of edges.
It is possible to tell whether a network is traceable or not without actually trying to trace it. Euler
discovered this in his attempt to solve the problem of the Konigsberg bridges (which he proved was
impossible). The key lies in knowing how many odd or even nodes the network contains. Why? Consider a
given network that is traceable and consider one of its nodes that is neither the start nor the end of a
journey through the network. Think of this – to get to such a node you must travel to it (since you aren’t
starting there). This means there is one edge needed for the trip to the node. Now, since you are not
finishing at this node, you must leave it – this requires another edge to leave by. Is such a node always odd?
Always even? Or does it vary?
If the beginning node and the finishing node are the same node, then it is an even node. We determined in
the previous paragraph that all nodes that were not the beginning or finishing node would have to be even
nodes. In conclusion, all the nodes of a “type 1” network are even and any one of them could be the beginning
and/or finishing node(s).
If the beginning node is different from the finishing node, they would both have to be odd or both have to
be even. (If they are both even, it falls in Type 1 above.) If they are both odd and, as we’ve previously
determined, all other points are even, to draw this network, we start at one odd node and end at the other.
In summary:
1) A network is traceable on if it contains no or exactly two odd nodes.
2) If all the nodes of a network are even, a traceable trip may begin and end at any node.
3) If a network contains exactly two odd nodes, it is traceable and the journey must begin at one of the
odd nodes and end at the other.
Determine if each of the following networks is traceable and complete. If it is traceable, identify the
beginning and ending nodes.
1.
2.




6.


8. 







5.
3.



 
 
 
 
 
 
 

9.
















4.





7.












 
  

10. 
11.










12.
13.
14.
15.
16.
17.
The Seven Bridges of Konigsberg
In 1735, Leonhard Euler used networks to solve a famous problem about the bridges in the city of
Koenigsberg, which is now Kaliningrad. The center of this city was on an island in the middle of the river
Pregel. The island was connected by four bridges to the banks of the river and by a fifth bridge to another
island, which was joined to the rest of the city by two more bridges. The people of Konigsberg wondered if
it were possible to travel through the city and cross each of the seven bridges only once each. After many
had tried it and had been unsuccessful, most people decided that it couldn’t be done, but they didn’t know
why.
Leonhard Euler, a great Swiss mathematician, was able to prove that it couldn’t be solve. This attempt
developed the theory of networks.
The method Euler used to solve the Konigsberg bridge
question was much like that we just used in learning about
networks. He began by utilizing what was to become a
property of topology – he reduced the land masses to
points (nodes) and stretched the bridges to edges thereby
forming a network.
A
Land A
1
2
Land B
3
5
4
Land D
1
6
2
B
Land C
A
6
1
5
7
4
3
D
7
C
B
2
4
3
6
5
C
7
D
Now using the properties of networks that we just developed, you can see that the network resulting from
the Konigsberg bridge map contains four odd nodes making it impossible to travel in a traceable fashion. The
only solution Euler could find for the people of Konigsberg was to put an eighth bridge between any two of
the four land masses. This would turn two of the four odd nodes into even nodes and it would be a network
of type 2.
Applications
Today, the study of networks and related theory has been expanded leading to many applications. Managers
have networks that represent both time and interrelationships of the many resources scheduled for a
project; and for proper planning, organization, and optimization of resources, they need to know the longest
and shortest paths through that network. In the area of communications, both the telephone company and
large corporations using leased telephone lines are concerned about the shortest paths and least-cost paths
from one point (city) to another. A manufacturer may want to know where to locate new plants and
warehouses in order to minimize the cost of shipping raw materials to plants and finished products from
plants to warehouses, wholesalers, and customers. The theory can also be applied to study processes in
chemistry, electronics, and aeronautics. Most problems solved by network theory involve so many points and
such volumes of data that their solution would not be practical without the use of computers.
An entertaining application of network theory similar to the Konigsberg bridge puzzle is these floor plans.
The challenge is to go through each door of the plan only once. Treat the room as Euler did the land masses
(nodes) in his solution to the Konigsberg bridge question and doors as he did the bridges (edges).
In each of the following floor plans, determine if it is possible to go through each of the doors only once. If
it is, determine if you must begin and end in certain rooms (identify those rooms). Note: The “outside” is
considered a room.
(a)
(c)
(b)
(d)
Topology or Rubber Sheet Geometry
Topology is a branch of mathematics that deals with the ways in which figures can be
distorted by stretching, shrinking, twisting, or bending without changing certain basic
properties. When considering plane figures, the distance between any two points can
change and the area enclosed within the figure can change, but the order of the points and
the way the points are connected cannot change. In other words the figure cannot be
broken or connected at any new points. (In topology, this is called homeomorphism.)
Two sets of topologically equivalent plane figures are shown below:
Set 1
Set 2
Are the figures in Set 3 topologically equivalent? __________________
Set 3
I. Group the following figures into sets so that the members of each set are topologically
equivalent to each other. Hint: There are seven different sets – one set has only one
member.
A.
B.
C.
D.
E.
Set 1: _____________________
Set 2: _____________________
F.
G.
H.
I.
Set 3: _____________________
Set 4: _____________________
Set 5: _____________________
J.
K.
L.
M.
Set 6: _____________________
N.
O.
P.
Set 7: _____________________
II. Group these forms of letters shown here to form topologically equivalent sets of figures.
Hint: There are nine sets.
Topologically equivalent solids are classified into categories called genus according
to how many holes are in their mass. Their equivalency is often illustrated by transforming a
doughnut made of modeling clay into a coffee cup. (This is also referred to homotopy
classes.)
A doughnut and a coffee cup have genus 1.
A baseball and a sugar cube have genus 0.
A figure 8 and this sugar
bowl have genus 2.
I (Doughnut)
II (Pill)
III (Wrench)
III. Choose I, II, or III as the figure to which each is topologically equivalent.
a) ball
b) pencil
c) scissors
d) needle
e) sugar bowl
f) ice cube
g) drinking glass
h) spool of thread
i) die
j) battery
k) funnel
l) brick
IV. In each exercise 1-5, tell which figure is not topologically equivalent to the rest:
1.
a.
b.
c.
d.
2.
a.
b.
c.
d.
3.
a. a solid ball
b. a hollow ball
c. a teaspoon
d. a comb
4.
a. a saucer
b. a car key
c. a coffee cup
d. a wedding ring
5.
a. a hammer
b. a screwdriver
c. a thimble
d. a sewing needle
IV. Group the block number below into three groups such that the numbers within each
group have the same genus.
0 1 2 3 4 5 6 7 8 9