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David Greenhood, Mapping,
Chapter 1, 2, 3
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Side Notes |
Links |
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Arcs and angles use the same units of measurement |
These
are the key concepts in this lesson. You should be able to define or
describe them after you have read this lesson, finished the assigned
reading and explored the links. You should also be able to do the
exercises and answer the questions at the end of the lesson.
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Arc length is different from arc measure
Converting from DMS to decimal form
Relationship between north-south distance and latitude
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Arc lengthThe units of length for an arc are linear units, e.g. meters, feet, kilometers. For a given central angle, larger circles correspond to greater arc lengths. To establish a relationship between measure of a central angle and corresponding arc length, we use an established relationship between the circumference of any circle and the diameter of that circle:Circumference / diameter = Pi or(Pi is approximately 3.142, or 22/7) Circumference = Pi * diameterSince that ratio of circumference to diameter applies to all circles, it is helpful in making a formula that relates the DMS measure of an angle or arc to the length of the arc. Since, for any circle, Diameter = 2 * radius,We have Circumference = 2 * radius * PiOn any circle, the length of an arc is proportional to its measure, e.g. doubling the arc measure causes doubling of the arc length. And in particular, (arc length)/circumference = (arc measure)/360And so Arc length = circumference * (arc measure)/360 = 2 * Pi * radius * (arc measure)/360 = radius * (arc measure) * (Pi/180)where arc measure is in the DMS system. For calculation relating angle measure to arc length and radius, we need first to express the DMS quantity in terms of degrees only because the formulas are in terms of degrees only. To do this we will convert minutes and seconds to fractions of a degree. Remember that one degree = 60 minutes, and one minute = 60 seconds. So one second = 1/60 minute, and one minute = 1/60 degreeThen, for example, to determine the length of an arc whose measure is 23° 15' 47", and which is part of a circle whose radius is 25 centimeters, we first write the measure in terms of degrees only: 23° 15' 47" = 23° + 15 * 1/60 degrees+ 47 * 1/60 minutes = 23° + 15 * 1/60 degrees+ 47 * 1/60 * 1/60 degrees = (23 + 15/60 + 47/(60*60))°which is approximately 23.263° Then Arc length = radius * (arc measure) * (Pi/180) = 25 * 23.263 * Pi/180which is approximately 25 * 23.263 * 3.142/180or approximately 10.152Since the radius is measured in centimeters, the arc length is approximately 10.152 centimetersRelating this formula to Earth measure, the distance from the equator to a point at latitude 37° N can be calculated as follows: Distance = arc length = (radius of Earth) * (arc measure) * (Pi/180)which is approximately (6370 kilometers) * (37 degrees) * 3.142/(180 degrees)or approximately 4114.1 kilometersSince in any circle the measure of any arc is the same as the measure of its central angle, we have a similar formula in terms of the central angle rather than the arc: Arc length = radius * (central angle measure) * (Pi/180)We have many occasions to measure length, on a plane or on a sphere. The examples above are intended to display rather straightforward relationships among arc length, arc measure and angle measure. In particular, many applications of Earth measure involve relationships between arc length and arc measure, i.e. relationships between distance and latitude or longitude. |
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Points on the same great circle are "collinear"
Sides of triangles are arcs of great circles. |
AreaAs in a plane, calculation of area on a sphere, especially for irregular shapes, is more difficult than is the case for distances along lines and other paths. In a plane, irregular shapes can be described or approximated in terms of mathematical functions; in other cases, methods for estimating area involve approximating the irregular shape as a polygon and calculating the area of the polygon.Spherical triangle On a sphere, we will use the spherical "triangle" as a primitive shape. A spherical triangle is defined by three points (vertices) that are not "collinear", i.e. not on the same great circle. The "sides" of the spherical triangle are great circle arcs whose endpoints are the vertices of the triangle. A general formula for areas of spherical triangles involves spherical trigonometry. Some special cases can be addressed through the formula for area of a sphere in terms of its radius: Area of surface of a sphere = 4 * Pi * (radius)^2 (Note: 4 * Pi * (radius)^2 = 4 * Pi * radius * radius)Then, for example, using 6370 kilometers as the radius of Earth, we can calculate the area of the Western Hemisphere. The area of the Western Hemisphere is half of the area of Earth. So Area of the Western Hemisphere = (1/2) * 4 * Pi * (6370)^2 = 2 * Pi * (6370)^2which is approximately 2 * 3.142 * 6370 * 6370 square kilometers or 254,985,000 square kilometersAs another example, note that for the spherical triangle whose vertices are (0°N 45°W), (0°N 45°E) and the North Polethe spherical triangle occupies one-eighth of the surface of Earth, so its area is approximately (1/8) * 4 * Pi * (6370)^2 square kilometerswhich is approximately 63,746,000 square kilometers. |
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Paths of constant direction
Strategies for determining elevation |
Rhumb linesOn Earth's surface, paths of constant direction are called "rhumb lines", or "loxodromes." In particular, travel along meridians is in constant direction, i.e. either due north or due south. And similarly, travel along parallels is in constant direction, either due east or due west. Other paths of constant direction are neither parallels nor great circles. For example, travel from a point on the Equator and bearing precisely north east, i.e. at an angle 45° with both due east and due north, will intersect each meridian at a 45° angle. Such a path will spiral through the Northern Hemisphere toward the North Pole.Mercator Projection One commonly-used projection
of Earth to a plane is the Mercator projection. Such a projection, i.e.
from a sphere to a plane, necessarily involves some distortion. The
Mercator projection involves distortions of distance and shape, but it
facilitates use of paths of constant direction. On a Mercator map, rhumb
lines are actually line segments. Thus, for example, a line segment on
the map from New York to London will describe a path that is in fact not
a line or a great circle, but a path that intersects meridians at the
same angle measure. Before the use of electronic aids for navigation
across oceans, sailing along a rhumb line, by use of stars or magnetic
compass, was an important option toward reaching one's destination. Earth as spheroidIn sufficiently small regions, e.g. for a city map, Earth measure can be approximated as geometry of a plane. In larger regions, distances, areas and angle measures are accurate enough if Earth is regarded as a sphere. For still more accuracy, we work with approximations that reflect Earth's deviation from spherical shape. Hence the use of 6370 kilometers or 3960 miles only as approximations for an average radius of Earth. Earth's radius is greatest at the Equator, least at the poles. Estimated range is from 6357 km. to 6378 km. or from 3951 miles to 3964 miles.Among other problems, irregularities in Earth's shape make difficult
the determination of altitude. Traditional use of barometric pressure is
subject to inaccuracies caused by atmospheric change. Use of electronic
models aids in addressing problems in measurement caused by
irregularities in Earth's shape. One strategy is the representation of
portions of Earth's surface as parts of a spheroid, in this case a
surface of revolution of an ellipse around its minor axis. There are
many such full-scale models and they are used locally in measurement of
altitude. Another strategy is the use of contours of equal gravitational
attraction, called "geoids", which fill the role of mean sea
level in Earth measure and, beyond that, bring us closer to the
relationship between Earth's geometric center and its gravitational
center. Other coordinate systemsWhile the latitude-longitude (lat-lon) system has much historical support and is visually easy to comprehend, it also has an important disadvantage. The relationship between longitude and east-west distance varies according to latitude. For example, at the equator the arc length between 0° longitude and 30° E longitude is greater than the arc length between those longitudes at 45° N latitude. One system designed to address this problem is the Universal Transverse Mercator projection. Its coordinate system is organized in terms of distances from the equator and from reference meridians. Units of measure for location and for distance are meters. Thus there is no unit conversion from map coordinates to linear units. While the complexity of the projection produces an unappealing image, application involving distances and areas on Earth's surface produce more consistency in results of calculation and of accuracy at various latitudes. Links Angle measure: postulates and theorems. Dictionary of units of measurement. Erathosenes' calculation of diameter of Earth
Exercises
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