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David Greenhood, Mapping,
Chapters 1 and 2
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Side Notes |
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Paths of travel along Earth
Great circle distance |
We view Earth here as a sphere. In fact there are surface irregularities, e.g. mountains and canyons, and its overall shape is a slight distortion of a sphere. But on an accurate model of manageable size, e.g. the size of a terrestrial globe on a desk, the irregularities are barely observable, if at all. Thus, in addressing need for a system for naming every location on Earth, for navigating on the surface, for planning routes of travel, and other applications, we begin with the geometry of the sphere. Spherical geometryAs described in Session 2, the surface of Earth is a two-dimensional entity. Any point on the surface can be named uniquely with two coordinates. To develop a coordinate system, we organize a collection of coordinate lines. But since our surface is a sphere, the coordinate lines are circles. The relative stability of the Earth's spinning establishes the location of the North Pole and South Pole as endpoints of the diameter of the sphere that contains the axis of rotation and that serve as anchors for the latitude-longitude coordinate system. All paths of travel along the spherical surface are curved. The shortest path of travel between two points is an arc of a circle that contains the two points and whose center is the center of the sphere. This is called a "great circle" of the sphere. While there are many circular paths connecting the two points, in general only one of them, the great circle, contains the path of minimum length. Exceptions to this are pairs of points that are at the ends of a diameter. Then all circles containing such a pair are great circles and all circular paths between the points contain minimum paths. In fact, all circular paths between two such points are minimum paths. The distance between two points is the length of the shorter arc connecting the points and contained in a great circle. For any two points that are not endpoints of a diameter, there is only one such circle. Length is represented in common linear units, e.g. meters, miles, feet, millimeters, etc. Two-dimensional measure: As in a plane, the common two-dimensional measure on a sphere is area. The relationship between shapes of a sphere and a plane is more difficult to address computationally than is the relationship between a circular arc and a line because the trigonometry of the sphere is considerably more complicated than the trigonometry of the circle. In practice, area on a sphere is calculated through estimation except in special cases for which there are formulas that produce exact areas. |
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Location
Names for meridians and parallels |
Latitude
and longitude: In most contexts, the traditional and pervasive
coordinate system is the latitude-longitude system. Places are named in
terms of semi-circles whose endpoints are the North and South Poles -
these semi-circles are called "meridians" - of the circle that
intersects the meridians midway between the poles - this is called the
"equator" - and of circles on the sphere that are parallel to
the equator - these are called "parallels". Directions north and
south are parallel to meridians and are marked by parallels. Directions
east and west are parallel to the equator and the parallels, and are
marked by meridians.
From any point on the sphere except for the poles, travel due north is along a unique meridian toward the North Pole; travel due south is along a unique meridian toward the South Pole. From the South Pole, all travel is due north; from the North Pole, all travel is due south. From any point on the sphere except for the poles, travel due east or due west is along a parallel. Such a route never reaches a pole. To use the system of meridians and parallels for assigning names to points on the sphere, we assign names to each meridian and each parallel. To give names to meridians, we give names to their points of intersection with the equator. Since the equator is a circle, we use units of circular measure, traditionally degrees, minutes and seconds. The equator is divided into 360 intervals of equal length, called degrees; each degree is divided into 60 intervals of equal length, called minutes; each minute is divided into 60 intervals of equal length, called seconds; seconds are subdivided into decimal fractions. This degree-minute-second system is abbreviated as the "DMS" system. The units are represented by symbols as in the following example: 25 degrees + 17 minutes + 38.4 secondsis represented as 25° 17' 38.4" |
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Using longitude for east-west measure
Using latitude for north-south measure |
Longitude:
For historical reasons, marking of the equator starts with the meridian
that contains Greenwich, England. Marking proceeds eastward (positive) and
westward (negative) from that. The degree-minute-second value of a point
on the equator is its longitude. All points on the meridian between that
point and the North Pole and between that point and the South Pole have
the same longitude. For example, the longitude of the city hall of San
Francisco, California, and all points due north or due south, is
approximately 122°25' west, or -122°25'. The longitude of Greenwich is
zero degrees.
With north-south variation, the east-west distance between meridians varies. Because distances between the meridians varies, a relationship between distance and longitude is defined for the equator: One minute of arc on the equator Remember that the arc measure of the equator is 360 degrees and
one degree of arc is 60 minutes of arc. So the arc measure of the
Equator is
With the poles as an exception, every point on the sphere is located with two coordinates: latitude and longitude. For example, the city hall of San Francisco is at approximately 37° 46' north latitude and 122° 25' west longitudewhich is commonly written as 37° 46' N 122° 25' WThe North and South Pole have unique latitudes: 90° north and 90°south, respectively, and no other point has either of these latitudes. Since they do not have unique longitudes - all meridians intersect at the poles - locations of the poles are named only by their latitudes. Hemispheres: Besides using DMS to name locations, there are commonly named portions of the sphere. Among these are the hemispheres. All points north of the equator comprise the Northern Hemisphere; all points south of the equator comprise the Southern Hemisphere. The great circle containing the Greenwich meridian also contains the International Dateline, with some deviation from the meridian for political integrity. The International Dateline intersects the Equator at 180° longitude, east or west. All points with west longitude comprise the Western Hemisphere; all points with east longitude comprise the Eastern Hemisphere. |
Discussion of latitude, longitude, meridian, parallel.
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Orbital geometryThe orbit of Earth around the sun is elliptical, very nearly circular, and it lies in a plane. The earth spins as it travels in its orbit and its axis maintains a constant direction. With respect to the plane of the orbit, the earth's axis makes an angle of approximately 66.5°. So it makes an angle of approximately 23.5° with a perpendicular to the plane of the orbit. While the axis maintains constant direction, it is also closely aligned with Polaris, i.e. the North Star. For that reason, from any point in the Northern Hemisphere at any time of the year, Polaris appears in the north at an angle of elevation whose measure is the same as the latitude of the point.Seasons: During the portion of Earth's orbit when axis is tilted, at its constant angle, with the North Pole toward the sun and South Pole away from the sun, the Northern Hemisphere is in summer and the Southern Hemisphere is in winter. Then half-way through the orbit when the axis tilts with the South Pole toward the sun, the Southern Hemisphere is in summer and the Northern Hemisphere is in winter. Spring and autumn, i.e. transitions between winters and summers, occur when the Earth is mid-way in its orbit between winter and summer. Points on Earth where the sun is directly overhead at some time during the year are between 23.5° N, i.e. the Tropic of Cancer, and 23.5° S, the Tropic of Capricorn. The sun is directly over the Tropic of Cancer on approximately (usually) June 21. This time is the summer solstice in the Northern Hemisphere and the winter solstice in the Southern Hemisphere. The sun is directly over the Tropic of Capricorn on approximately (usually) December 21. This time is the summer solstice in the Southern Hemisphere and the winter solstice in the Northern Hemisphere. The sun is directly over the Equator on approximately March 21 - this time is the vernal equinox in the Northern Hemisphere and the autumnal equinox in the Southern Hemisphere - and on approximately September 21 - this time is the autumnal equinox in the Northern Hemisphere and the vernal equinox in the Southern Hemisphere. |
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Zones:
The band around the earth between the Tropics is called the Torrid Zone.
Points where the sun does not rise for at least one day of the year are
between 66.5° N latitude and the North Pole, or between 66.5° S and the
South Pole. Points north of 66.5° N comprise the Arctic Zone; points
south of 66.5° S comprise the Antarctic Zone. Points between the Tropic
of Cancer and the Arctic Circle comprise the North Temperate Zone, and
Tropic of Capricorn and the Antarctic Circle comprise the South Temperate
Zone.
Calendar: A day on Earth is approximately the time of one revolution of Earth on its axis. It is the span between two consecutive times when the Sun, with respect to some observation point, is at the same direction, e.g. from noon to noon, 5PM to 5PM. There are approximately 365 days in one year. More accurately, but not exactly, there are approximately 365 1/4 days in one year, hence leap years, i.e. years whose names on the Gregorian calendar are multiples of four. In leap year, February has an extra day, i.e. February 29, to compensate for the effect of underestimating each year by 1/4 day. To refine again the measure of a year, leap years occur in years whose names are multiples of 400, e.g. 2000, but not in other years whose name is a multiple of 100, i.e. 1900. As Earth rotates from west to east, i.e. so that the sun rises in the east, one boundary between consecutive days remains fixed: the International Date Line. The other boundary, where it is midnight, moves from east to west. Regardless of time of day, the region immediately west of the International Date Line is at one day later than is the region to the east. And the region immediately east of the meridian where the time is midnight is at one day later then is the region west of the meridian. |
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Links Diagram showing traditional special days based on Earth orbit. Lunar year and the Chinese calendar Number theoretical aspects of lunar and solar years
Exercises
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