The Haversine Method



Geometry and IGA-1

[pic]

All North and East coordinates are POSITIVE Prime Meridian

All South and West coordinates are NEGATIVE

(-, +) (+,+)

All coordinates must also be in RADIANS

instead of Degrees. A circle contains 360 degrees Equator

or [pic]radians, so 180 degrees = [pic]radians

(-, -) (+,-)

From your notes worksheet you completed earlier today, we must change the coordinates to +/- and radians.

|City |Latitude with N or S|Longitude with E or|Latitude in Degrees |Longitude in Degrees|Latitude in Radians |Longitude in Radians|

| |(degrees) |W | | |(multiply by |(multiply by |

| | |(degrees) |(+/- instead of N or|(+/- instead of E or|[pic]) |[pic]) |

| | | |S) |W) | | |

|Columbus, Ohio |40.0 N |83.0W |40 |-83 |.6981 |-1.4486 |

|Sydney, Australia | | | | | | |

|London, England | | | | | | |

|Mexico City, Mexico | | | | | | |

|Rio de Janeiro, | | | | | | |

|Brazil | | | | | | |

|Cape town | | | | | | |

|South Africa | | | | | | |

To find the distance between 2 cities: City1: (latitude, longititude)

City2: (latitude, longititude)

The Haversine Formula is:

[pic]

[pic]

[pic], Your calculator must be in Radians not degrees!!!!

[pic] (if [pic]see Mr. Reese)

Distance, [pic] where R is the earth’s radius, so [pic]

Example: Find the distance between Columbus Ohio and Sydney Australia:

|City |Latitude with N or S|Longitude with E or|Latitude in Degrees |Longitude in Degrees|Latitude in Radians |Longitude in Radians|

| |(degrees) |W |(+/- instead of N or|(+/- instead of E or|(multiply by |(multiply by |

| | |(degrees) |S) |W) |[pic]) |[pic]) |

|Columbus, Ohio |40.0 N |83.0W |40 |-83 |.6981 |-1.4486 |

|Sydney, Australia |33.8S |151.2 E |-33.8 |151.2 |-.5899 |2.6389 |

City1: Columbus, Ohio: (.6981,-1.4486) [pic] = [pic] = 4.0875

City2: Sydney Australia: (-.5899, 2.6389) [pic]= [pic]= -1.288

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic] [pic] = [pic] = [pic] = [pic]

Distance, [pic] where R is the earth’s radius, so [pic] [pic]

1. Please note how the above example was done in an organized, easy to follow manner.

Together, as a class we will now use the Haversine Formula and find the distance between Columbus, Ohio and Mexico City, Mexico.

HW. Assignment: use the Havesine Formula and find the distance for any 2 of the following:

a)Columbus to London, England b)Columbus to Cape town South Africa c)Columbus to Rio de Janeiro Brazil

Make sure your work is done in an organized, easy to follow manner. We will be using the website listed on your Culture Fair Requirements sheet to check your distance.

[pic] Geometry and IGA-1 Requirement

Your group is investigating a certain country or culture. Your trifold board must include some form of a map indicating where the “focus city” of your country or culture is located.

Your trifold board must also list the Geographic Coordinates of this city.

You will list these as Latitude________________ N or S, Longitude ____________ E or W

You must then find the distance, in miles, from this city to Columbus, Ohio using the Havesine Formula. All work must be shown in an organized, easy to follow manner.

Your group will also find the distance from Columbus to your city by entering the coordinates of Columbus, Ohio and your city into the website:

You must enter the coordinates as: ddd.ffffD

where ddd = 0-3 digits, ffff = 0-10 digits and D = N,S,E, or W. This format represents a decimal number of degrees. If the number of degrees is a whole number, the decimal point is optional.

For Example, Columbus Ohio; the source city would be entered as: 40.0N (Lat) 83.0W (Long)

Sydney Australia; the destination city would be entered as 33.8S (Lat) 151.2E (Long)

After entering the coordinates of your city in the “destination” section, leave the units in statute miles and submit your query.

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Mathematics using Latitude and Longitude. Finding the distance between locations.

The distance formula [pic] is based on the Pythagorean Theorem and is valid for plane (flat) surfaces.

Since the world is spherical we must use a more mathematically involved formula, the Haversine Formula:

Name__________

Make sure your calculator is in radians!

To “test” sin 30 = -.98803 (calc. in radians)

sin 30 = .5 (if calc. is in degrees)

Round off to 4 decimal places.

See Culture Fair Geo. & IGA-1 Requirements

For the website that calculates distances between geographical coordinates. The distance is using this Haversine formula is withine ½ mile of the actual distance!!!!

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