mapping mathematics

8/10/2019 Mapping Mathematics

1/16

Mapping Mathematics

NCTM Southern Regional ConferenceCharleston, SC

November 6, 2003

Dr. David RoysterCenter for Mathematics, Science, and Technology Education

UNC CharlotteCharlotte, NC 28223

[email protected]


2/16

Mapping Mathematics

NCTM Southern Regional Conference 2003 2 Dr. David Royster

CMSTE, UNC Charlotte

History

Early world maps reflect the religious beliefs of the form of the

world. For example maps have been discovered on Babylonian

clay tablets dating from around 600 B.C. One such map showsBabylon and the surrounding area in a stylized form with

Babylon represented by a rectangle and the Euphrates river by

vertical lines. The area shown is depicted as circular surroundedby water which fits the religious image of the world in which the

Babylonians believed.

Eratosthenes, around 250 B.C., made major contributions tocartography. He measured the circumference of the Earth with

great accuracy. He sketched, quite precisely, the route of the Nile

to Khartoum, showing the two Ethiopian tributaries. He made

another important contribution in using a grid to locate positionsof places on the Earth. He was not the first to use such a grid for Dicaearchus, a follower of

Aristotle, had devised one about 50 years earlier. Today we use latitude and longitude to

determine such coordinates and Eratosthenes' grid was of a similar nature. Note, of course, thatthe use of such positional grids are an early form of Cartesian geometry. Following Dicaearchus,

Eratosthenes chose a line through Rhodes and the Pillars of Hercules (present day Gibraltar) to

form one of the principal lines of his grid. This line is, to a quite high degree of accuracy, 36north and Eratosthenes chose it since it divided the world as he knew it into two fairly equal parts

and defined the longest east-west extent known. He chose a defining line for the north-south

lines of his grid through Rhodes and drew seven parallel lines to each of his defining lines to

form a rectangular grid.


3/16

Mapping Mathematics



Hipparchus was critical of the grid defined by Eratosthenes, saying reasonably enough that it was

chosen arbitrarily. He suggested that a grid should be chosen with astronomical significance sothat, for example, points on the same line would all have the same length of longest day.

Although Hipparchus never constructed a map as far as we know, he did make astronomical

observations to describe eleven parallels given by his astronomical definition. Although no

copies of the work by Eratosthenes and Hipparchus survives, we know of it through theGeographical Sketchesof Strabo which was completed in about 23 A.D. Although Strabo gives a

critical account of earlier contributions to cartography, he devotes only a small discussion to the

problem of projecting a sphere onto a plane. He states clearly that his work is not aimed atmathematicians, rather at statesmen who need to know about the customs of the people and the

natural resources of the land.

The final ancient Greek contribution we consider was the most important and, unlike that ofStrabo, was written by a noted mathematician. In about 140 A.D. Ptolemy wrote his major work

Guide to Geography, in eight books, which attempted to map the known world giving

coordinates of the major places in terms of what are essentially latitude and longitude. The first

volume gives the basic principles of cartography and considers the problem of map projection,that is mapping the sphere onto the plane. He gave two examples of projections, and also

described the construction of globes. Right at the beginning Ptolemy identifies two distinct types

of cartography, the first being:-

... an imitation through drawing of the entire known part of the world together with the things

which are, broadly speaking, connected with it.

The second type is:-

... an independent discipline [which]sets out the individual localities.

Now the main part of Geography consisted of maps but Ptolemy knew that although a scribecould copy a text fairly accurately, there was little chance that maps could be successfully

copied. He therefore ensured that the work contained the data and the information necessary for

someone to redraw the maps. He followed previous cartographers in dividing the circle of theequator into 360 and took the equator as the basis for the north-south coordinate system. Thus

the line of latitude through Rhodes and the Pillars of Hercules (present day Gibraltar) was 36

and this line divided the world as Ptolemy knew it fairly equally into two. The problem ofdefining lines of longitude is more difficult. It required the choice of an arbitrary zero but it also

required a knowledge of the circumference of the Earth in order to have degrees correspond

correctly to distance. Ptolemy chose the Fortune Islands (which we believe are the Canary

Islands) as longitude zero since it was the most western point known to him. He then marked off

where the lines of longitude crossed the parallel of Rhodes, taking 400 stadia per degree.


4/16

Mapping Mathematics



Ptolemy's map of the world.

Had Ptolemy taken the value of the circumference of the Earth worked out by Eratosthenes then

his coordinates would have been very accurate. However he used the later value computed by

Posidonius around 100 B.C. which, although computer using the correct mathematical theory, is

highly inaccurate. Therefore instead of the Mediterranean covering 42 as it should, it covers 62in Ptolemy's coordinates. Books 2 to 7 of Geography contain the coordinates of about 8000

places, but although he knew the correct mathematical theory to compute such coordinates

accurately from astronomical observations, there were only a handful of places for which suchinformation existed. It is not surprising that the maps given by Ptolemy were quite inaccurate in

many places for he could not be expected to do more than use the available data and, for

anything outside the Roman Empire, this was of very poor quality with even some parts of the

Roman Empire severely distorted.

At a time when Christian Europe was producing religious representations of the world rather

than scientific maps, another type of map, or perhaps more accurately chart, for the use of sailors

began to appear. These were calledportolanmaps (from the Italian word for a sailing manual)and were produced by sailors using a magnetic compass. The earliest examples we know about

date from the beginning of the 14th

century, and were Italian or Catalan portolan maps. The

earliest portolan maps covered the Mediterranean and Black Sea and showed wind directions andsuch information useful to sailors. The coast lines shown on these maps are by far the most

accurate to have been produced up to that time. The Catalan World Map produced in 1375 was

the work of Abraham Cresques from Palma in Majorca. He was a skilled cartographer andinstrument maker and the map was commissioned by Charles V of France. The western part of

his map was partly based on portolan maps while the eastern part was based on Ptolemy's data.


5/16

Mapping Mathematics



Mercator made many new maps and globes, but his greatest contribution to cartography must be

theMercator projection. He realized that sailors incorrectly assumed that following a particular

compass course would have them travel in a straight line. A ship sailing towards the same point

of the compass would follow a curve called a loxodrome (also called a rhumb line or sphericalhelix), a curve which Pedro Nunez, a mathematician greatly admired by Mercator, had studied

shortly before. A new globe which Gerardus Mercator produced in 1541 was the first to have

rhumb lines shown on it. This work was an important stage in his developing the idea of theMercator projection which he first used in 1569 for a wall map of the world on 18 separate

sheets. The 'Mercator projection' has the property that lines of longitude, lines of latitude and

rhomb lines all appear as straight lines on the map. In this projection the meridians are verticaland parallels having increased spacing in proportion to the secant of the latitude. Edward Wright

published mathematical tables to be used in calculating Mercator's projection in 1599.

Mercators World Map A Mercator Map of the Americas


6/16

Mapping Mathematics



Latitude and Longitude

DEFINITIONS:

Latitude

Measurement of distance in degrees north or south of the Equator; from the Latin latus,meaning wide.

Longitude

Measurement of distance in degrees east or west of the prime meridian; from the Latin

longus, meaning length.

WHERE ARE YOU?

The Earth's circumference is divided into 360 degrees (you should keep a world globe

handy when introducing these topics), each degree represented by an imaginary line

running either east-west or north-south on the surface of the Earth. The prime meridian isthe imaginary line which runs through the geographic North Pole to the geographic South

Pole, passing through Greenwich England. The prime meridian is defined as 0 E and 0

W. Each meridian (longitude line) to the west of Greenwich is labeled 1-180 W, whileeach meridian to the east of Greenwich is labeled 1-180 E.

Why was Greenwich, England selected as 0E and 0W?(Find out on the Internet!)

Since the Earth's circumference (a complete circle) is 360 degrees, then directly opposite

the prime meridian is a meridian which can be labeled both as 180 E and 180 W. Thismeridian is called the International Date Line. Likewise, the Equator is 0 N and 0 S

latitude. The geographic North Pole is 90 N latitude, while the South Pole is 90 S

latitude.

Why is 180E and 180W labeled the International Date Line?

Therefore, if you are trying to describe to anyone exactly where you're located on the

planet Earth, latitude and longitude will provide you with a means of answering the

question Where are you?.


7/16

Copyright 1988-2002 Microsoft Corp. and/or its suppliers. All rights reserved. http://www.microsoft.com/streets Copyright 2001 by Geographic Data Technology, Inc. All rights reserved. 2001 Navigation Technologies. All rights reserved. This data includes information taken with permission from Canadian author

Charleston, SC

0 mi 0.2 0.4


8/16

The Bar (graphic) Scale is particularly important when enlarging or reducing maps

by photocopy techniques because it changes with the map. If the Bar Scale is

included in the photocopy, you will have an indication of the new scale


9/16

Mercator Projection

USGS: http://lessonplanet.teacherwebtools.com/search/redirect?lpid=1175&mfcat=/search/Geography/Maps/startat10/&mfcount=67

ake a ercator Projection

Follow the directions below to make a close approximation of the normal Mercator projection. A few activities with the mapare included to demonstrate the important characteristics of this projection.

Time

One 50-minute period for steps 1-10.

After photocopying in step 11, one 50-minute period for steps 11-14.

aterials needed per person

Protractor Compass

Ruler

Two sheets of 11 x 17 inch paper

Transparent tape

Sharp pencil

Fine point pen, preferably black

Procedures

1. Tape the two pages together along the I7-inch sides and orient the paper as shown in illustration A, with the tapeon the reverse side.

2. The joint between the two pieces of paper will be the "equator." Lay the protractor on the paper with the flat sideon the left. Place the zero point and the 90-degree mark on the equator at least 2 inches from the edge of thepaper. Use the compass to draw a semicircle with a 6-inch diameter, flat side on the left. Mark the center (the zeropoint, Z) on the diameter. IMPORTANT NOTE: Make all marks lightly in pencil, unless otherwise instructed.

3. Using a protractor, mark every 10 degrees around the semicircle (see illustration C). Starting at the top, label thesepoints A, B, C, ..., S.

s

a bd

f

hz


10/16

Mercator Projection


4. Beginning at Z, measure left along the equator 2/5 of a radius (in this case, 1.2, inches) and mark a new point, T,as shown above.

5. Using the protractor, draw the westernmost line of longitude perpendicular to the equator and tangent to theoriginal semicircle at point J.

6.

Set the spacing of the lines of latitude as follows: With the left end of your ruler on point T, align the right side topoint I on the semicircle; mark where this line (TI) intersects the westernmost longitude line. Beginning again atpoint T, mark points on the westernmost longitude for lines through points H, G, F, E, D, and C. Each point marks10 degrees of latitude.

7. Draw latitude lines parallel to the equator through these new points. To make the latitudes parallel, measure thedistances between marks on the westernmost longitude line; copy these measurements and mark equivalent pointson the easternmost longitude line. Connect pairs of points (a western and an eastern), preferably beginning closestto the equator. What would be a more geometric way of constructing these latitude lines?

8. Repeat steps 6 and 7 for latitudes south of the equator. Notice that on this projection, lines of latitude are paralleland spacing between them increases away from the equator. Latitudes 90N and 90S cannot be shown on aMercator projection, because they are infinitely far from the equator (although this approximate construction doesnot show this).

9. Set longitude lines as follows: Measure east 0.5 inches from the westernmost longitude and make a mark on theequator. This represents 10 degrees of longitude. Repeat this step 17 more times, and you will have 180 degrees oflongitude. From each point on the equator, use the protractor to draw a perpendicular line. On the Mercatorprojection, longitude lines are parallel and equally spaced, as shown in illustration G.

10. At this point, the map covers only half the planet (a hemisphere). Carefully trace this grid in ink.

zs

s

zT J

60

60

0

180 W


11/16

Mercator Projection


11. To map the entire Earth, make two copies of the original and join the copies along the one's easternmost line oflongitude and the other's westernmost (see illustration H). Lines should connect across the copies. These two linesboth represent the Prime Meridian, the line of 0 degrees longitude.

12. Label the latitude and longitude lines along the right and bottom of the map. The equator is 0 degrees latitude,

and latitude values increase in increments of ten to the north and south. The westernmost longitude line is 180degrees W; longitude values decrease in increments of ten to 0 degrees at the Prime Meridian, and increase againto 180 degrees at the eastern edge of the map (see illustration H). This map approximates the characteristics of theMercator projection within about 2 percent.

13. Make a bar scale in the margin below the map. A bar scale is commonly centered below the map, in this case,below the Prime Meridian. To determine the scale at the equator, divide the Earth's equatorial circumference(24,902 miles; 40,075 kilometers) by 360 degrees; therefore, each degree of longitude and latitude at the equatorequals about 69 mi (about 111 km). Ten degrees of longitude at the equator (about 690 miles) is represented by 0.5inch on the map; one inch represents 1,380 miles. Draw a line representing 3,000 miles (about 5,000 kilometers).

14. Locate your home city on the map by longitude and latitude. Locate at least 5 of the following cities: London, Paris,Moscow, Washington, Cairo, Buenos Aires, Sydney, Beijing, Calcutta, Kabul, Baghdad, Johannesburg. Find thecompass headings from your home town to each of the cities that you chose.

180 W180 E

60 N

60 S

0

180 W 180 E0

1500 mi0


12/16

Map Projections

Make a map of the Earth

1. Cut off the top of a soda bottle where the bottle straightens. Cut off the skinny neck of the

bottle without slicing through the "globe."

2. Draw lines of latitude and longitude on the inside of the globe. Doing this will be easier using

a bowl or some other round object to guide the marker.

3. To represent a planar projection, place the globe on a piece of tracing paper. Hold theflashlight above the globe and shine its light down through the globe and onto the tracing paper.

Draw the lines of latitude and longitude as they are projected on the paper.


13/16

4. For a cylindrical projection, roll the tracing paper into a cylinder the diameter of the globe.

Shine the flashlight through the globe and onto the tracing paper. Draw the lines of latitude and

longitude as they are projected onto the paper.

5. A conical projection is a compromise between planar and cylindrical

projections. Make a cone out of the tracing paper and rest it on top of the

globe.

6. Shine the flashlight through the globe and into the cone. Draw the lines

of latitude and longitude as they are projected onto the cone.

7. Discuss alternative map-making methods that would minimize the

distortions inherent in maps.

Materials

Clear plastic soda bottle, flashlight, tracing paper, scissors, marker, pen

Last modified 06/02/2000 16:32:42

Copyright (c) 2000. Gulf of Maine Aquarium.

All rights reserved. Mail comments to [email protected]


14/16

Make Your Own Globe

Flat maps may be easier to carry around, but there is still a need to make globes so that Earth's geography

can be viewed without any directional or spatial distortions. Printing the location of continents and oceans

directly onto a round surface would be difficult. Instead, this map of the Earth is printed in flat, roughlytriangular sections and then attached to a ball. These sections are called gores.

Make a globe1. Using a tape measure, determine the circumference of the ball, making sure that the tape measure

circles the ball without wandering away from the equator.

2. On a large piece of paper draw a rectangle the same length as the circumference of the ball. The

height of the rectangle should be half the circumference of the ball. Draw an equator line through

the center of the rectangle, lengthwise.

3. Cut out the rectangle.

4.

Place the rectangle in front of you horizontally. Fold it in half three times. Unfold the rectangle

and there are eight equal sections. Draw a line along each fold. Measure the bottom edge of onesection to find its midpoint, and mark that point A. Mark the end of the equator in that end

section B.

5. Find the midpoint between A and B as follows: Place the compass point on A. Set the compass

radius to a length just short of B and draw a semicircle. Maintaining the same radius, place the


15/16

compass point on B and draw a second semicircle. The two semicircles should intersect at two

points. Draw a straight line through the points where the semicircles intersect, extending the line

to a point at which it intersects the equator line. Mark this point C.

The length of the line from A to C is the radius of the gores.

6. Attach extra paper to both ends of the original piece (These extensions should be at least the

length of the gore radius) Extend the equator line out onto the extra paper at least the distance ofthe gore radius. This will allow you to move your compass point out along the equator far enough

to draw all of the gores.

7. Set the compass to the gore radius (the distance between A and C). Place the compass pencil on A

and the compass point on C. Draw an arc from A to the top of the rectangle.

Maintaining the same compass radius, move the compass pencil to the midpoint of the bottom

edge of the next section and place the compass point on the equator. Draw another arc in the same

manner. Continue moving the compass and drawing arcs for each of the eight sections.

Turn the paper upside down and repeat the above procedure to draw the opposing arcs and formthe gores.

Remove the extra paper from the rectangle.

8. Create grid lines for transposing the map onto the rectangle as follows :

Fold the rectangle as you did in Step 3. Fold once more in the same direction. Unfold the

rectangle, and place it in front of you horizontally. Fold it in half, top to bottom, three times.

Unfold the rectangle. The rectangle should be divided into 16 sections left to right and 8 sections

top to bottom. Cut out the spaces between the gores. Transpose the map from the given to your

gores using the gridlines on the diagram and the gridlines (folds) on the gores as a reference.

9.

Tape the strip of gores at one end of the equator to the ball. Wrap the strip around the ball and

tape the loose ends in place, taking care to align the equator line.

10.Glue each gore down against the ball so the tips meet to form the north and south poles.


16/16

Make an Orange Globe

To demonstrate the difficulty in making a flat map of a round surface make a map out of an orange peel.

Try to peel an orange with an X-acto knife so that you take off the skin in one piece. Make a flat

projection of the surface of the orange by laying the peel flat. Does your peel look like the gores from the

previous activity?

Materials

paper, ball (about the size of a playground ball), glue, tape, paints, ruler, tape measure, compass, pencil

Last modified 06/02/2000 16:32:50Copyright (c) 2000. Gulf of Maine Aquarium.

All rights reserved. Mail comments to [email protected]

mapping mathematics

Documents