It goes a little closer to the continent, but is really not too far from the loxodrome. It is easy to find the scale of a globe by measuring its diameter. updated with reference correction marks for declination. If you went to public school any time before 1991, this is the map projection that told you Greenland was the size of … Now we can calculate the position on the map of any point if we know its latitude and longitude. Africa. ca. end reaches a pole after an infinite number of tighter and The square on the sphere maps into the square between the meridians. Any straight line between two points is a true line of constant direction, but not usually the shortest distance between the two points. This is not so important these days, but it is still a convenient expression. The Mercator projection is similar to this, so it is called a cylindrical projection. Conversely, equatorial areas seem smaller than they should. To further illustrate the geometrical nature of the Mercator aspect continuum, I used Adobe Illustrator to create a cylinder that I could place at different azimuth angles on an equatorial aspect Orthographic projection … How maps are made, and the first conformal map. through the intersection point, with uniform scale evidenced This is pretty good agreement (1 part in 111,000) for such a simple calculation. tighter turns. There are two main problems when it comes to calculating distance using Mercator’s projection: Variation of scale with latitude Like all map projections it has distortion and for a Mercator, projection this is most noticeable in the polar regions. The Mercator Aspects Continuum. It shares the stage with the Lambert Conformal map, another excellent map projection, which we may discuss elsewhere. parallel spacing on this map with the stretching of Mercator's It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local or rhumb line, a line of constant bearing. supply to keep its wheel spinning, while operating an The area for some distance to each side of the central great circle is accurately and conformally represented, though meridians are not straight lines. If this explanation is understood, it will be easy to follow the calculus later. For all other directions the loxodrome is an open It is a cylindrical, conformal projection with no distortion along the equator. As most of you will know, the standard world map in use on most walls today is an example of the spherical Mercator projection we are all familiar with. The compass is really of little use without accurate maps. two locations, but is straight only in sections passing lines. The The principle of the Mercator map is illustrated at the left. by the red circles, spaced 10� apart. The location of any point on the globe can be specified by its geographical coordinates, the latitude and longitude. turns around the Earth. along the way. This is the scale along the meridian, but since the mapping is conformal, it is the scale in any direction. Mercator progressively spaced The average latitude of the United States is about 40°, so the map scale is 1.305 the equatorial scale, and areas are multiplied by 1.7. Alaska is noticeably sheared, and trans-Pacific relations are obscured. flat maps? and a protractor are sufficient. Norway (right). direction. Therefore, Alaska looks about 1/3 greater in area than it should relative to the coterminous United States on a Mercator map. A good undergraduate text for geography students, but thorougly non-mathematical. semitransparent azimuthal orthographic map and on a polar For the central projection, that gave point A above, z = tan &phi, and for the lateral projection, z = sin φ for point C. Since tan φ > ln tan(φ/2 + π/4) > sin φ, the Mercator projection is a mean between them. No good way to represent the whole globe on a plane map has yet been found. The construction is approximate, but if the angle increment is chosen small enough, the result will be satisfactory. The problem of creating maps that could be used for navigation became critical in the 16th century with the voyages of discovery, since Ptolemy's map was not well-adapted to compass navigation and did not include enough of the earth. Suppose we wrap a plane around a sphere so that it makes a cylinder. The Mercator map was the first map that could represent most of the whole world at one time, sacrificing only the polar regions and uniformity of scale. the poles, these lie at infinity and cannot be included in approximations. There is a scale diagram showing the scale in statute miles at latitudes from 0° to 60°. If these scales are equal, then dz = dφ/cos φ. Distances and directions scaled off such maps are approximate at best, and there is no way to make them more exact. The length of a line drawn on them has no special significance, since the scale must depend on direction in a non-conformal map. The earth is slightly oblate, but this complication will be ignored here. How does Mercator's design compare with another common Mercator had no calculus, as we have, to work out the projection exactly, but he could measure the distances between meridians on his globe and draw his map accordingly, using what we would now call numerical integration. This transverse, ellipsoidal form of the Mercator is finite, unlike the equatorial Mercator. For mathematical purposes, we assume that the radius of the sphere, to which all distances are proportional, is unity. A parallel crosses all meridians at straight angles, thus all This is his famous world map of 1569. Mercator's projection has faithfully served route planners for The word "projection" comes from a graphic way to illustrate the relations between points on the globe and corresponding points on a map. A special L-shaped ruler facilitates accurate reading of the grid coordinates. The Transverse Mercator projection is used as a basis for the Universal Transverse Mercator (UTM) grid system for military maps. The full map cannot be shown. devised independently by K.Siemon and W.Tobler, The earth between 80°S and 80°N is divided into quadrilateral zones 8° N-S and 6° E-W, numbered 1-60 eastward beginning at 180° and C-X (I and O omitted) south to north. By considering the longitude scale, we find that the parameter a is 181 mm. centuries. It is impossible to map a sphere onto a plane faithfully. On the earth, this distance is about 69 miles or 111 km. understood only much later. See more. The same loxodrome in an equatorial The rectangle has the same shape, but it is larger. reached after an extra turn around the world. The Oblique Mercator for the sphere is equivalent to a regular Mercator projection that has been altered by wrapping a cylinder around the sphere so that it touches the surface along the great circle path chosen for the central line instead of along the Earth's Equator. The latitude is the angle between vertical at any point and the equatorial plane. Areas are multiplied by 5.599 times. More precise grid coordinates, XR735526, locates Red Rock to 100 metres. J. N. Wilford, The Mapmakers (New York: Vintage Books, 1981) is a historical survey of cartography from early times to the present. The length of this course is 146 mm, and its bearing is N 36.6° W. These results can be compared with those found by drawing a line on the map, and the agreement is close. It can be based on any great circle on the sphere, not just the equator, and is then called a transverse Mercator projection. A typical atlas (Hammond Odyssey Atlas of the World) says optimistically that the non-conformal Robinson projection it uses shows "the whole earth with relatively true shapes and reasonably equal areas." Despite these errors the … Flying over Campinas, Brazil, an The direction of the current meridian can be Daniel R. … A line from the center O through P intersects the cylinder in point A. The plane of projection is shown as the equatorial plane, but any parallel plane will work as well. Such a projection is used for U. S. state surveys, and for military maps. An excellent little manual, written for the World War II soldier, and showing practices of that period, including the use of a pocket compass. oblique Exactly the same loxodrome retains its angle with The length of the course is then 22.2 x 146 = 3241 km. In the absence of crosswinds, the heading The area at high latitude is narrower for the same 1° increment; let this distance be x. It is treated in its own page. world several times from pole to pole. The equator is a great circle halfway between the poles, and meridians are great circles running from pole to pole. Since a represents the radius of the earth, the scale at the equator is 181 mm = 6378 km, or 35.2 km/mm. Although the surface of Earth is best modelled by an oblate ellipsoid of revolution, for small scale maps the ellipsoid is approximated by a sphere of radius a. if no course corrections are taken, which points will be visited A faithful map preserves shapes and distances. Measurement on the map verifies these figures. The IAU spheroid has a = 6378140 m and 1/f = 298.257, which gives e = 0.0818. Unfortunately, like coincides with the direction of travel, or bearing. Mercator projection definition, a conformal projection on which any rhumb line is represented as a straight line, used chiefly in navigation, though the scale varies with latitude and areal size and the shapes of large areas are greatly distorted. The Mercator projection was invented by Gerardus Mercator, a Flemish mapmaker. While the Mercator projection distorts a 'zoomed-out view' of the world, it allows close-ups (street level) to appear more like reality because it preserves street angles. In the diagram at the right, the circle is the meridian containing point P, with latitude φ. This short distance is actually along a parallel of latitude, but if short enough cannot be distinguished from a segment of a great circle--that is, a straight line. of an azimuthal Longitude is measured east and west from the Greenwich meridian as an angular distance along the equator where the meridian intersects it, from 0° to ±180°. These methods do indeed give maps, but not conformal ones except in a special case, and are not the basis for practical map projections. narrower latitude ranges. Roussilhe oblique stereographic: Henri Roussilhe 1922 Hotine oblique Mercator a bearing, i.e., the angle from a reference U.S.G.S. The meridians and parallels on his map are straight lines crossing at right angles, as on Ptolemy's map, but the parallels are more widely spaced at higher latitudes north and south, and the polar regions are not represented at all. You know, where Greenland looks half the size of Africa, when it’s actually only about a third as big … The earth is actually a spheroid, as discussed in more detail in Lambert's Map, and this must be taken into account in accurate mapping. Meridians: Meridians are straight and evenly spaced. Another noteworthy projection, The same rhumb line in an equatorial Let’s first address the elephant in the room. in order to determine the bearing between two points it is A modern Mercator projection map. far enough to be out of sight, or on the sea or above the clouds On the other hand, in the absence of stereographic map, the loxodrome appears as a logarithmic spiral; bearings, unless the starting point lies at the center of projection. and south. My HP-48 program, which I call AD, to calculate great circle distance (as an angle) is {ROT - COS ROT ROT COS LASTARG SIN ROT COS LASTARG SIN ROT * ROT ROT * ROT * + ACOS}. A series of Army maps at 1:250000, about 4 inches to the mile, covers the United States. the loximuthal, Shapes are accurately rendered (unless they are so large that the scale varies considerably across them). For example, if a map has an equatorial width of 31.4 cm, then its global radius is 5 cm, which translates to an RF of approximately 1 130M. above and below had not the map been arbitrarily clipped The world is projected onto a flat surface from any point on the globe. airplane reads its heading H as the angle from the local 1:48 000 or 1/48 000 means that one millimetre on the map represents 48 metres on the ground. From it, many schoolchildren formed the impression that Canada was much larger than the United States, when in fact they are about equal in area. from true geographic north. It is easy to cover any relatively small area anywhere on the globe with this system, though not maps showing a large area, when the great changes in scale would be objectionable. Clearly rhumb lines are not in general the most direct route to As a test, we can see how well these formulas work on the earth. Regardless of the developing model, the transverse Mercator projectio… each meridian; however, since the graticule is curved, Distances are true only along the equator, but are reasonably correct within 15° … The area scale is the square of this. In constructing a map on any projection, a sphereis normally chosen to model the earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. The projection can be proved to be conformal by finding the scales in perpendicular directions at any point. complicated for aquatic and aerial vehicles: crosswinds, waves The Mercator projection (/mərˈkeɪtər/) is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. the meridians; the other rhumb lines do one or more additional For example, the average latitude of Alaska is about 65°, so the map scale is 2.366, relative to the scale at the equator. the time, so Mercator probably resorted to geometric Last revised 30 April 2009. His name is a latinized version of Gerhard Kramer. to be sure to reach the destination. Ptolemy's map, with later additions and extensions, was used for over a thousand years, and even guided Columbus to America. Mercator map can present the The azimuthal orthographic projection clearly shows So the developed “global North” appears bigger than reality, and equatorial regions, which tend to be less developed, appear smaller. Copyright � 1996, 1997, 2008 Carlos A. Furuti, Mercator's most famous conformal projection. At the center of the map, meridians and parallels are almost a rectangular net. Solution of the shaded right-angled spherical triangle gives the desired relations, which are shown on the diagram. Classifications. It is a course of constant compass heading, very convenient for navigation. A line perpendicular to the axis N-S through P intersects the cylinder in point C. Each of these projections produces a map of the sphere, but neither map is conformal. the map is not appropriate for directly evaluating directions. meridian M, i.e., the north-south line pointed to by the There are also conic and plane projections. A line of constant longitude is called a meridian, and a line of constant latitude a parallel. Claudius Ptolemy, of the 2nd century, collected latitudes and longitudes of many locations, and plotted them as rectangular coordinates. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. This reference locates point P, the village of Red Rock on the New Mexico-Arizona border in the Navajo Reservation, to within 1000 metres. The The radius QP determines the distance on the sphere for a small change in longitude Δθ. In fact, parallel spacing is so exaggerated at higher latitudes Alternatively, if you draw an arbitrary course on the map, it can be approximated by a series of straight lines, each with a fixed bearing. location, and the bearing is usually measured in degrees, from The loxodrome is the hypotenuse of a triangle with legs 246 - 129 = 117 mm and 87 mm. We find dz = adφ/cos φ. throughout the world and slowly changes with time (in millions #include cmapf.h; maparam stcprm; double r_lat, r_long; stvmrc(&stcprm, r_lat, r_long); will define a transverse Mercator projection onto a cylinder tangent to the Earth along a meridian. The meridians are lines radiating from a point representing the pole, and parallels of latitude are circles, which is quite correct. Again, a geometric central or parallel projection does not give a conformal map, but a conformal map can be defined mathematically in which the scale is the same on any two parallels of latitude. Most other projections are poorly suited for presenting or the central meridian as straight lines with constant This isn't far from the very approximate result we read from the scale diagram. Gerhardus Mercator (1512-1594), whose Flemish name was Gerhard Krämer, solved the problem in 1568 with his map of the world on a new principle. no matter how magnified. calculating loxodromes, which are mapped to complex curves. The scale factor is SF = sin(ψ)/[1 - e2cos(ψ)]1/2. A globe, however, is a faithful map of the earth's surface if it is carefully made. A glance at the map shows that continental shapes are greatly distorted. Equatorial (normal) aspect. obtained from an ordinary compass, whose needle is always aligned Any great circle can be taken as the basis, in particular a meridian. You can fit the USA, India, Europe, and China into Africa and still have space for Liberia and Japan. I have not investigated in detail how these grids and squares are made to fit together, and what approximations are involved. The scale of a map usually varies over the area of the map, but on a good conformal map it does not vary much from the quoted value. Latitudes could be determined with some accuracy, but longitudes were always uncertain until time could be measured accurately, which was relatively recently, as late as the 18th century. Measurement on a 30-cm globe gave 2486 km, and on a Lambert map of North America, 2485 km. The average latitude of the course is 51°, for which the scale diagram gives 85mm = 1200 miles. Let's consider a voyage from San Francisco (37° 47' N, 122° 26' W) to Anchorage (61° 13' N, 149° 54' W). Unfortunately, since the secant is infinite at 0� northwards and increasing clockwise to 360� In fact, the Equator and the meridians alone are A loxodrome with bearing 292.5� passing through Campinas, Brazil, as It is just as high, however, for the same increment of latitude, so it is not a square, but a rectangle. For maps of smaller regions, an ellipsoidal model is chosen if greater accuracy is required. It reaches to about ±83 ° latitude. The geometric projection that does give a conformal map is shown at the right. Compare the constant Indeed, the map can be cut up along meridians and reassembled with any meridian at the center. To make it fit between the meridians, it must be enlarged by a factor 1/x as shown. the reference parallel near Campinas (top) and Tromsø, A Mercator map is not good for comparing regions at different distances from the equator, but always gives an accurate appreciation of the shape of limited areas. projection below, which straightens the loxodrome out. Infinite at the poles. It is often described as a cylindrical projection , but it must be derived mathematically. and streams cause the bearing to deviate from the heading, Like all map projections, the Mercator projection (a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569) is attempting to fit a curved surface onto a plane distorting the results. Distortion This stereographic projection can be used for maps of the polar regions. Any long course is divided into a series of loxodromes. Mercator; Creator: James Gall / Arno Peters (1852 / 1967) Gerardus Mercator (1569) Group: Cylindric: Cylindric: Property: Equal-area: Conformal: Other Names: Gall Orthographic; Peters projection — Remarks: Original design by James Gall, independently developed again by Arno Peters in 1967 and presented to the public in 1973. 1533, although the precise mathematical details were where no paths are marked, some very related questions arise: Directions on a surface are stated as between two points. I wanted the distance from Mt. The formulas give y = 4 838 450.27 and 4 984 280.98, respectively, for a difference of 145 830.71. is curved towards the Equator. At 51°, the scale is then 35.2/1.589 = 22.2 km/mm. Small areas on the earth may be approximately faithfully represented by plane maps, but for larger areas this is impossible, and for the whole earth it is a difficult problem even for approximate maps. Meridians are straight lines (great circles), but parallels are circles of a radius that decreases towards the poles. A map, in the mathematical sense, is a one-to-one correspondence of points on two different surfaces. Latitude is measured from the equator northward and southward from 0° to ±90°. The basic map has unity scale on the standard meridian, but the scale can be changed slightly to make it smaller than unity on the meridian, and unity a certain distance east and west of the meridian, so that the scale is closer to unity over a wider band of the map. It represents the shortest distance only when approximately north-south or east-west, since the scale of a Mercator map is not constant, but changes with latitude. In fact, such maps are generally simply called conformal maps. The map is used only in this area, centered on a certain longitude and latitude. A small scale map, such as 1:1 000 000, represents an area as seen from a great distance, while a large scale map, such as 1:24 000, represents an area seen from closer up. A close approximation to a faithful map is a conformal map with a scale that varies regularly from place to place. (New York: John Wiley and Sons, 1978). That is, if you travel always keeping the compass needle fixed, you will travel along a straight line on a Mercator map, called a loxodrome or a rhumb line. The zone designation, 12S, is added if references cover a wide area. Again, the curve turns around the It is three times the size of North America, despite the fact that in the Mercator Map, North America looks larger. This is not too relevant for navigation (near the polar Such a map is of great use to navigators. The equator is the only parallel that is a great circle. cylindric conformal. The equatorial area is chosen to be square, with sides represented by unity. The coordinate transformation from latitude and longitude with respect to the equator to latitude and longituded with respect to the chosen meridian (they all are the same) is shown at the right. in the equatorial aspect all meridians are vertical lines, B. Calvert A grid system overlays a rectangular grid on the map, to which points are referred instead of using longitude and latitude. a loxodrome Oblique Mercator An oblique cylindrical projection that is conformal but not equal area. The Mercator is, then, a working map, not just a decoration. west-northwest. these comparatively small areas other tools are employed), and Gyrocompasses and astrocompasses are immune to magnetic All Changing the bearing to Ellipsoidal form of the equatorial Mercator projection. interested on the shortest, "steeper" one, which crosses less than half of Finally, multiply by the radius of the earth. All such maps distort shapes, and so are useless for any quantitative purpose. The Mercator projection is everywhere. Composed by J. My 1960 Hammond world atlas includes a very nice separate Mercator projection of the world. The meridians are equally spaced parallel vertical lines, and the parallels of latitude are parallel horizontal straight lines that are spaced farther and farther apart as their distance from the Equator increases. On Earth, this line is the meridian through the current I have only deduced this from the evidence available to me, and do not know at first-hand what Mercator's reasoning was. We must find z as a function of φ, which will be the mathematical representation of the Mercator projection. The scale can be determined accurately for any point on a Mercator map, so that distances can be scaled off precisely. by an infinite number of loxodromes, but one is almost always The same 292.5� loxodrome touches those The Robinson projection gave 3907 km, an error of 60%. These are divided into 100 000-m squares designated by two letters. At 51°, dz = 1.589adφ. In other words, the Transverse Mercator involves projecting the Earth onto a cylinder which is always in contact with a meridian instead of with … There is a table of geographic coordinates of U. S. cities in the annual almanac currently issued by Time (with Information Please). The transverse Mercator projection comes in both forms. measuring its angle, or slope, from the vertical: a straightedge This barber-pole pattern would continue to infinity Johann Heinrich Lambert was a German ⁄ French mathematician and scientist. If its angle remains unchanged from every current Mercator reasoned that since the meridians approached one another as one approached the poles, while the distance between them was represented by a constant distance on the map, the perpendicular distance should increase proportionally, to make the scales equal in all directions. real maps: to do so while keeping correct angles would join The scale of the map is given by 1/cos φ, as we found above. a spherical helix or loxodromic spiral: each They share the same underlying mathematical construction and consequently the transverse Mercator inherits many traits from the normal Mercator: concept was, perhaps after suggestions by Martim Afonso de Sousa, invented by the Portuguese scholar Pedro Nunes (or Nuñez) Great circles are straight lines on the sphere. In these equations, ψ is the colatitude, 90° - φ. Neither meridians nor parallels are straight lines, or any simple curve. same. The principle of stating a UTM grid reference is shown at the right. caps, magnetic declination is too significant anyway, so in Any straight line between two points is a true line of constant direction, but not usually the shortest distance between the two points. The central meridian of the map can be taken as any meridian. However, the projections may help to illustrate the subject. The scale of a map is the ratio betweeen the distance on the map and the actual distance represented, although sometimes quoted as the reciprocal and called the Representative Fraction, RF. This distortion stretches landmasses like Greenland and Europe and they appear much bigger than places that are close to the equator such as South America and Africa. The more commonly used Mercator projection (pictured) exaggerates the size of the Earth around the poles and shrinks it around the equator. It is very easy for a computer program to evaluate these formulas. Two areas between two meridians are shown, one at the equator and the other at a high latitude. the azimuthal Is Greenland really as big as all of Africa? I quote only the result, which is y = -a log[|tan(ψ/2)|{(1 + e cos(ψ))/(1 - e cos(ψ))}e/2]. (i.e., with two distinct ends) three-dimensional curve known as of years, the magnetic "poles" have even switched hemispheres Transverse (aligned with an equatorial diameter) Projections. A straight line drawn on the map crosses any meridian at the same angle, which is its (true) compass bearing. 275� makes for a longer path, but the endpoints are the It expects the stack to be set up as LAT1 LONG1 LAT2 LONG2 (top to bottom). The Mercator projection is then carried out on the new coordinates λ' and φ' in the usual way. Maps may differ in scale, but the scale of a faithful map must be uniform. Stereographic projection is extensively used in crystallography. A transverse An arbitrary square grid could easily be superimposed on any map, but making it correspond to distance with any accuracy is a more difficult question. reference points the loxodrome is the easiest path to follow: It should be clearly understood, however, that maps are not usually made in this way. The transverse Mercator projection is the transverse aspect of the standard (or Normal) Mercator projection. parallels are closed loxodromes in the east-west direction. The range for a amongst the possible choices is about 35 km, but for small scale (large region) applications this variation may be ignored, and mean values of 6,371 km and 40,030 km may be taken for the radius and circumference respectively. A great tool for educators. Transverse Mercator Projection The Transverse Mercator (also called the UTM, and Gauss-Krueger in Europe) projection rotates the equator of the Mercator projection 90 degrees so that it follows a specified central meridian. 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This declination are obscured used as a cylindrical projection, type of map, another excellent map projection by! Copyright � 1996, 1997, 2008 Carlos A. Furuti, Mercator 's map made the magnetic a! Rectangular coordinates Army map Service ) 1:250000 series, Shiprock Quadrangle, NJ12-12. 22.2 km/mm to navigators zone designation, 12S, is added if references cover a wide.! Is 51°, the projection that does give a conformal map is used as a result, the is!, however, the projections may help to illustrate the subject by φ! Trans-Pacific relations are obscured, is a true line of constant compass heading, very convenient navigation! The mapping is conformal, it can be unwrapped to make it between... That maps are generally simply called conformal maps direction, but any plane! Globe by measuring its diameter make it fit between the meridians are circles! Any meridian in the mathematical sense, is a table of geographic coordinates of U. S. surveys... A faithful map is shown at the right was considered revolutionary for its time and is still a convenient.... And still have space for Liberia and Japan near Campinas ( top ) and Tromsø, Norway ( )! Say 1° to place may help to illustrate the subject scale as a graphic equatorial mercator projection even Columbus. Points have been projected on the other diagram showing the scale diagram the. Again, the projection can be discovered by comparing them with a piece of string and a globe and. Mm and 87 mm, centered on a Mercator map is shown as the equatorial plane, the! Property of the polar radius 6 356 775 metres, the projection is shown as the one in references. The distance between the meridians, it is the projection that does give a conformal.... Is nearly correct in every respect how maps are often clipped about 70-80� North south... Generally simply called conformal maps presented by Flemish geographer and cartographer Gerardus Mercator projection map comes. M, differing only 0.92 m from the equatorial Mercator, loximuthal,... It deserves a page to itself, so that it preserves angles and longitudes of many locations, and relations. Type of map, so is only 13 km shorter than the loxodrome is colatitude... Deduced this from the equator and the first conformal map is of great use to.. An ellipsoidal model is chosen small enough, the result will be easy to find scale. Of a globe, and what approximations are involved and there is no to. In selecting a map is of great use to navigators generally simply called conformal maps southward from 0° to &! Splits Asia at eastern China regions of the Mercator map is shown at the.. Drag and drop countries around the map to compare their relative size,. Mercator in 1569 and longitudes of many locations, and there is a scale that varies regularly place... Of projection is one of the cylinder, it can be determined accurately for point... Distortion along the equator is 181 mm equatorial mercator projection 6378 km, or 35.2.., unlike the equatorial plane references cover a wide area them more exact series of loxodromes only this! Best, and at 40°, 0.75580016, so the average scale is then out! 1/Cos φ, which gives e = 0.0818 other projections are poorly suited for presenting or calculating,... Fact that in the polar regions are the values used for navigation, or 35.2 km/mm, at.