NCGIA Core Curriculum in Geographic Information Science
URL: "http://www.ncgia.ucsb.edu/giscc/units/u014/u014_f.html"

# Unit 014 - Latitude and Longitude

by Anthony P. Kirvan, Department of Geography, University of Texas at Austin, USA

This section was edited by Kenneth Foote, Department of Geography, University of Texas Austin.

This unit is part of the NCGIA Core Curriculum in Geographic Information Science. These materials may be used for study, research, and education, but please credit the author, Anthony P. Kirvan, and the project, NCGIA Core Curriculum in GIScience. All commercial rights reserved. Copyright 1997 by Anthony P. Kirvan.

Your comments on these materials are welcome. A link to an evaluation form is provided at the end of this document.

### Topics covered in this unit

• This unit provides an overview of latitude and longitude, including:
• Earth rotation, the North and South Poles, and the Equator
• Parallels of latitude and meridians of longitude
• Determination of north or south position with latitude
• The use of longitude to determine east or west position
• The measurement of latitude and longitude with degrees, minutes, and seconds

### Learning Outcomes

• After learning the material covered in this unit, students should gain an appreciation for:
• The relationship between plane and earth coordinate geometries
• The importance of the earth's rotation and poles to measurement and point location
• The use of latitude and longitude to determine locations on the earth's surface
• The differences and relationships between latitude and longitude
• Using latitude and longitude to measure distances

### Instructors' Notes

Unit 014 - Latitude and Longitude

## 1. Frames of Reference

• Throughout history, many methods of keeping track of locations have been developed.
• Plane coordinate geometry was developed as an abstract frame of reference for flat surfaces.
• Once the earth's round, three-dimensional shape was accepted, a spherical coordinate system was created to determine locations around the world.

### 1.1. Plane Coordinate Geometry

• René Descartes' contributions to mathematics were developed into cartesian coordinate geometry.
• This is the familiar system of equally-spaced intersecting perpendicular lines in a single plane.
• The two principal axes are the horizontal (x) and the vertical (y) .
• Any point's position can be described with respect to its corresponding x and y values (x,y).
• Figure 1.  The position (5,3) is a unique location easily plotted on the cartesian plane.
• The position of one point relative to another can also be shown with the cartesian coordinate system.

## 2. Earth Coordinate Geometry

• The earth's spherical shape is more difficult to describe than a plane surface.
• Concepts from cartesian coordinate geometry have been incorporated into the earth's coordinate system.

### 2.1. Rotation of the Earth

• The spinning of the earth on its imaginary axis is called rotation.
• Aside from the cultural influences of rotation, this spinning also has a physical influence.
• The spinning has led to the creation of a system to determine points and directions on the sphere.
• The North and South poles represent the axis of spin and are fixed reference points.
• If the North Pole was extended, it would point to a fixed star, the North Star (Polaris).
• Any point on the earth's surface moves with the rotation and traces an imaginary curved line:
• Parallel of Latitude

### 2.2. The Equator

• If a plane bisected the earth midway between the axis of rotation and perpendicular to it, the intersection with the surface would form a circle.
• This unique circle is the equator.
• The equator is a fundamental reference line for measuring the position of points around the globe.
• Figure 2. The equator and the poles are the most important parts of the earth's coordinate system.

### 2.3. The Geographic Grid

• The spherical coordinate system with latitudes and longtitudes used for determining the locations of surface features.
• Parallels: east-west lines parallel to the equator.
• Meridians: north-south lines connecting the poles.
• Figure 3.  The Geographic Grid
• Parallels are constantly parallel, and meridians converge at the poles.
• Meridians and parallels always intersect at right angles.

### 2.3.1. Parallels of Latitude

• Parallels of latitude are all small circles, except for the equator.
• True east-west lines
• Always parallel
• Any two are always equal distances apart
• An infinite number can be created
• Parallels are related to the horizontal x-axes of the cartesian coordinate system.
• Figure 4. Parallels of Latitude

### 2.3.2. Meridians of Longitude

• Meridians of longitude are halves of great circles, connecting one pole to the other.
• All run in a true north-south direction
• Spaced farthest apart at the equator and converge to a point at the poles
• An infinite number can be created on a globe
• Meridians are similar to the vertical y-axes of the cartesian coordinate system.
• Meridians of Longitude - Figure 4.

### 2.3.3. Degrees, Minutes, and Seconds

• Angular measurement must be used in addition to simple plane geometry to specify location on the earth's surface.
• This is based on a sexagesimal scale:
• A circle has 360 degrees, 60 minutes per degree, and 60 seconds per minute.
• There are 3,600 seconds per degree.
• Example: 45° 33' 22" (45 degrees, 33 minutes, 22 seconds).
• It is often necessary to convert this conventional angular measurement into decimal degrees.
• To convert 45° 33' 22", first multiply 33 minutes by 60, which equals 1,980 seconds.
• Next add 22 seconds to 1,980: 2,002 total seconds.
• Now compute the ratio: 2,002/3,600 = 0.55.
• The earth rotates on its axis once every 24 hours, therefore:
• Any point moves through 360° a day, or 15° per hour.

### 2.3.4. Great and Small Circles

• A great circle is a circle formed by passing a plane though the exact center of a sphere.
• The largest circle that can be drawn on a sphere's surface.
• An infinite number of great circles can be drawn on a sphere.
• Great circles are used in the calculation of distance between two points on a sphere.
• A small circle is produced by passing a plane through any part of the sphere other than the center.

### 2.3.5. Loxodromes

• Arcs of great circles are very important to navigation since they represent the shortest route between two points.
• A loxodrome, or rhumb line, intersects each meridian at the same angle (constant compass bearing).
• Unfortunately, this route traced by a loxodrome is not the shortest distance.
• Maintaining a constant heading or azimuth traces a sprial on the globe called a loxodromic curve.
• To approximate the path of a great circle, which constantly changes azimuth, navigators plot courses along a series of loxodromes.
• The Mercator projection was developed especially for navigators, and presents straight lines as loxodromes.
• Because of the great distortion of parallels and meridians on this projection, great circles appear as deformed curves.
• Figure 6 - Loxodrome and Great Circle Comparison on Mercator Projection

## 3. Using Latitude and Longitude for Location

### 3.1.  Latitude

• Authalic Latitude is based on a spherical earth:
• Measures the position of a point on the earth's surface in terms of the angular distance between the equator and the poles.
• Indicates how far north or south of the equator a particular point is situated.
• North latitude: all points north of the equator in the northern hemisphere
• South latitude: all points south of the equator in the southern hemisphere
• Latitude is measured in angular degrees from 0° at the equator to 90° at either of the poles.
• A point in the northern hemisphere 40 degrees north of the equator is labeled Lat. 40° N.
• Forty degrees south of the equator, the label changes to Lat. 40° S.
• The north or south measurement of latitude is actually measured along the meridian which passes through that location
• It is known as an arc of the meridian.
• Geodetic Latitude is based on an ellipsoidal earth:
• The ellipsoid is a more accurate representation of the earth than a sphere since it accounts for polar flattening.
• Refer to the Shape of the Earth (Unit 015) for more information.
• Modern large-scale mapping, GIS, and GPS technology all require the higher accuracy of an ellipsoidal reference surface.
• Refer to the Coordinate Systems Overview (Unit 013) for more information.
• When the earth's shape is based on the WGS 84 Ellipsoid:
• The length of 1° of latitude is not the same everywhere as it is on the sphere.
• At the equator, 1° of latitude is 110.57 kilometers (68.7 miles).
• At the poles, 1° of latitude is 111.69 kilometers (69.4 miles).
• Table 1 - Length of a Degree of Geodetic Latitude

### 3.1.1. Latitude and Distance

• Parallels of latitude decrease in length with increasing latitude.
• Mathematical expression:  length of parallel at latitude x = (cosine of x) * (length of equator)
• The length of each degree is obtained by dividing the length of that parallel by 360.
• Example: the cosine of 60° is 0.5, so the length of the parallel at that latitude is one half the length of the equator.
• Since the variation in lengths of degrees of latitude varies by only 1.13 kilometers (0.7 mile), the standard figure of 111.325 kilometers (69.172 miles) can be used.
• For example, anywhere on the earth, the length represented by 3° of latitude is (3 * 111.325)  333.975 kilometers.

### 3.2. Longitude

• Longitude measures the position of a point on the earth's surface east or west from a specific meridian, the prime meridian.
• The longitude of a place is the arc, measured in degrees along a parallel of latitude from the prime meridian.
• The most widely accepted prime meridian is based on the Bureau International de l'Heure (BIH) Zero Meridian:
• Defined by the longitudes of many BIH stations around the world..
• Passes through the old Royal Observatory in Greenwich, England.
• The prime meridian has the angular designation of 0° longitude.
• All other points are measured with respect to their position east or west of this meridian.
• Longitude ranges from 0° to 180°, either east or west.
• Since the placement of a prime meridian is arbitrary, other countries have often used their own.
• For the purposes of measurement, no one prime meridian is better than another
• Having a widely accepted meridian allows comparison between maps published in different areas.
• The distance represented by a degree of longitude varies upon where it is measured.
• The length of a degree of longitude along a meridian is not constant because of polar flattening.
• At the equator, the approximate length is determined by dividing the earth's circumference (24,900 miles) by 360 degrees: 111.05 kilometers (69 miles).
• The meridians converge at the poles, and the distance represented by one degree decreases.
• At 60° N latitude, one degree of longitude is equal to about 55.52 kilometers (34.5 miles).

### 3.2.1. Longitude and Distance

• Because the earth is not a perfect sphere, the equatorial circumfererence does not equal that of the meridians.
• On a perfect sphere, each meridian of longitude equals one-half the circumference of the sphere.
• The length of each degree is equal to the circumference divided by 360.
• Each degree is equal to every other degree.
• Measurement along meridians of longitude accounts for the earth's polar flattening:.
• Degree lengths along meridians are not constant:
• 111.325 kilometers (69.172 miles) per degree at the equator
• 16.85 kilometers (10.47 miles) per degree at 80° North
• 0 kilometers at the poles
• The distance between meridians of longitude on a sphere is a function of latitude:
• Mathematical expression: Length of a degree of longitude = cos (latitude) * 111.325 kilometers
• Example: 1° of longitude at 40° N = cos (40°) * 111.325
• Since the cosine of 40° is 0.7660, the length of one degree is 85.28 kilometers.
• Table 2 - Length of a Degree of Geodetic Longitude
• These lengths are based on an ellipsoid and are similar to the lengths computed with the spherical formula.

### 3.3. Calculating Distances in Latitude and Longitude

• Calculating distance on a sphere based on latitude and longitude is a complicated task.
• The calulation of the distance between two points on a plane surface is a relatively simple task and has promoted the use of two-dimensional maps throughout history.
• When calculating distances over large areas, the authalic sphere can be used as a reference surface.
• The shortest distance between two points on a sphere is the arc on the surface directly above the true straight line.
• The arc is based on a great circle.
• Table 3 - Great Circle Distance Calculation
• See the Web References section for online examples.
• The difference betwen the sphere and ellipsoid is important when working with large areas.
• At a scale of 1:40,000,000, a 23 kilometer error in distance would equal a pen line (0.5 mm) on paper.
• Complex geodetic models based on ellipsoids are necessary for precise meaurement.
• Loran-C requires range computations with better than 10 meter accuracy over 2,000 kilometers.
• Geodetic measurements using satellites requires very accurate range computations.
• Table 4 - Ellipsoidal Distance Calculation

## 4. Review and Study Questions

### 4.1. Essay and Short Answer Questions

• Describe the relationship between the cartesian coordinate system and the geographic grid.
• Convert 35.40° into degrees, minutes, and seconds.
• Explain why the length of a degree of longitude decreases as one approaches the poles.

### 4.2. Multiple-Choice Questions

• The equator can also be called a:
1. Prime Meridian
2. Parallel of Latitude
3. Great Circle
4. Both 1 and 2
5. Both 2 and 3
• Which of the following is not true of parallels of latitude?
1. They are true east-west lines
2. Any two are always equal distances apart
3. Always meet at the poles
4. Related to the x-axis of the cartesian coordinate system
• Which of the following is not true of meridians of longitude?
1. They always meet at the poles
2. True north-south lines
3. Each is equal to half the length of a great circle
4. Always begin with the Prime Meridian through Greenwich, England

## 5. Reference Materials

### 5.1. Print References

• Dent, Borden D.  Cartography: Thematic Map Design, William C. Brown Publishers, 1990.
• Muehrcke, Phillip C., and Juliana O. Muehrcke.  Map Use: Reading-Analysis-Interpretation, 4th ed. Madison, WI: J.P. Publications, 1998.
• Robinson, Arthur H., et al.  Elements of Cartography, New York: John Wiley & Sons, 1995.
• Strahler, Arthur N., and Alan H. Strahler.  Elements of Physical Geography: Fouth Edition, New York: John Wiley & Sons, 1989.
• Strahler, Arthur N., and Alan H. Strahler.  Modern Physical Geography: Third Edition, New York: John Wiley & Sons, 1987.

## Evaluation

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## Citation

To reference this material use the appropriate variation of the following format:
Kirvan, Anthony P. (1997) Latitude/Longitude, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u014/u014.html, posted (today).