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

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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

Full Table of Contents

Metadata and Revision History

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.
  • Adding this to 45 degrees, the answer is 45.55°.

• 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.

• Figure 5 - Great and Small Circles

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.
    • Long range radio navigation requires precise distances.
    • 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.

5.2. Web References

• Bali Online: How far is it? - This service calculates the distance between two places.

• Geographic Names Information System (GNIS) - This database can be used to locate the longitude and latitude of over 1,233,933 features in the U.S.

• GeoSystems Information on Latitude and Longitude

• Bureau of the Census Gazetteer - Search for latitude and longitude data by entering a zip code or town and state information.

Evaluation

Citation

Kirvan, Anthony P. (1997) Latitude/Longitude, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u014/u014.html, posted (today).