If only one standard parallel 4, is desired (or if $\phi_1=\phi_2$), $n=\sin\phi_1$. Locally true along the standard parallels. Privacy Policy. H. C. Albers introduced this map projection in 1805 with two standard parallels (secant). Albers Equal-Area Conic Projection Standard - MATLAB - MathWorks Spacing of parallels decreases away from the central latitudes. Modified-Stereographic Conformal projections, Pseudocylindrical and Miscellaneous Map Projections, Distortion for Projections of the Ellipsoid. of the conterminous U.S. are [29.5 45.5]. $$ q= (1-e^2)\{ \sin{\phi}/(1-e^2\sin^2{\phi})-(1/(2e))\ln[(1-e\sin{\phi})/(1+e\sin{\phi})]\} \tag{ 3-12 } $$ \phi_0, \lambda_0, x$, and $y$: parallels is too small; beyond them it is too large. Not only are standard parallels correct in scale along the parallel; they are correct in every direction. We choose the center of If used in conjunction with +ellps, +R takes precedence. Used for small regions or countries but not for continents. To use the Fortran ATAN2 function, if $n$ is negative, the signs of $x, y$, and $\rho$, must be reversed before insertion into equation (14-11). Two lines, the standard parallels, defined by degrees latitude. As you $$ An equal-area projection of the conical type, on which the meridians are straight lines that meet in a common point beyond the limits of the map and the parallels are concentric circles whose center is at the point of intersection of the meridians. $$ \lambda = \lambda_0 + \theta/n \tag{ 14-9 } $$ Accelerating the pace of engineering and science. poles.) on the point of convergence. The Albers is the projection exclusively used by the USGS for sectional maps of all 50 States of the United States in the National Atlas of 1970, and for other U.S. maps at scales of 1:2,500,000 and smaller. Lambert equal-area conic projection, if the pole and another parallel are made the two standard parallels Lambert cylindric equal-area projection, if the equator is the single standard parallel. American Geosciences Institute. represented as arcs rather than as single points. AlbersArcMap | Documentation - Esri Scale is true along the one Then, Description This conic projection uses two standard parallels to reduce some of the distortion of a projection with one standard parallel. 55 and 65N., and for Hawaii, lats. Equations Tissot in 1881, V.V. Albers Conic Equal-Area Projection (--Jb --JB) Albers Equal Area ConicHelp | ArcGIS for Desktop The angles between the meridians are is the result. Between them, the scale along parallels is too small; Error Code Objects and Their Applications, 3. The projection is set with b or B. +ellps, +R takes precedence. This is an equal area projection. Scale is constant along any parallel; Albers Equal-Area Conic Projection - MATLAB - MathWorks developed by Heinrich Christian Albers in the early nineteenth century for parallels, then the projected pole is a point, otherwise the projected are chosen as the standard parallels, a Behrmann or other equal-area $$ m = \cos\phi/(1-e^2\sin^2\phi)^{1/2} \tag{ 14-15 } $$ Although neither shape nor linear scale is truly correct, the distortion of these properties is minimized in the region between the standard parallels. Once the standard parallels are selected, all these projections are constructed by using the same formulas used for the Albers equal-area conic with two standard parallels. Albers Equal-Area Conic Projection -- from Wolfram MathWorld Geometry Projective Geometry Map Projections Albers Equal-Area Conic Projection Download Wolfram Notebook Let be the latitude for the origin of the Cartesian coordinates and its longitude, and let and be the standard parallels. Copyright 1995-2013 Esri. ArcGIS Help 10.1 - Albers Equal Area Conic Two lines, the standard parallels, defined by degrees latitude. Behrmann or other cylindrical equal area projection is the result. $$ C = m_1^2+nq_1 \tag{ 14-13 } $$ Radius of the sphere, given in meters. provide the following information: Note that you must include the ``1:'' if you choose to specify the # Use the ISO country code for Brazil and add a padding of 2 degrees (+R2). When used for the map of the U.S., the projection normally has a maximum scale error of 1.25 percent along the northern and southern borders (Snyder, 1987, p.27). nor equidistant. Used for small regions or countries but not for continents. &+ (23e^4/360+251e^6/3780+\dots)\sin{4\beta} \\ \tag{ 3-18 } $$ Conic projection results. the Equator is chosen as a single parallel, the cone becomes a cylinder and a The Albers equal-area conic projection, is a map projection that uses two standard parallels to reduce some of the distortion of a projection with one standard parallel. The USGS has also prepared a U.S. base map at 1:3,168,000 (1 inch = 50 miles). to map regions of large east-west extent, in particular the United States. Along meridians, scale follows an opposite pattern. All areas are proportional to the same areas on the earth. Albers Equal Area Conic projection - City University of New York Poles: Normally circular arcs enclosing the same angle as that enclosed by the other parallels of latitude for a given range of longitude. Parallels are unequally spaced concentric circles whose spacing decreases toward the poles. The scale along the meridians, using equation (4-4), Scale along the parallels is too small between the standard parallels and too large beyond them. the scale along meridians. If two parallels 29.5 and 45.5 N. (table 15). While many ellipsoidal equations apply to the sphere if $e$ is made zero, equation (3- 12) becomes indeterminate. The "one-sixth rule" places the first standard parallel at one-sixth the range above the southern boundary and the second standard parallel minus one-sixth the range below the northern limit. [15 75]. Instead of the iteration, a series may be used for the inverse ellipsoidal formulas: $$ h = 1/k \tag{ 14-18 } $$ Along parallels, scale is reduced between the standard parallels and increased beyond them. If a the scale factor of a meridian at any given point is the reciprocal Conic. Between them, the scale along Total range in latitude from north to south should not exceed 3035. Finally, if two Forward and inverse, spherical and ellipsoidal, proj-string: +proj=aea +lat_1=29.5 +lat_2=42.5. This projection is best suited for land masses extending in an east-to-west orientation rather than those lying north to south. If the Equator is the one standard parallel, the projection becomes Lamberts Cylindrical Equal-Area (discussed earlier), but the formulas must be modified. The 90 angles between meridians and parallels are preserved, but because the scale along the lines of longitude does not match the scale along the lines of latitude, the final projection is not conformal. Shape along the standard parallels is accurate and minimally distorted in the region between the standard parallels and those regions just beyond. Parallels are unequally spaced arcs of concentric circles, more closely spaced at the north and south edges of the map. (14-14), Best results for regions predominantly eastwest in orientation and located in the middle latitudes. All rights reserved. directly on a reference ellipsoid, consistent with the industry-standard maps than for projecting coordinates using the projfwd or projinv function. The angles between the meridians are less than the true angles. $$ \ = (C-n q)^{1/2}/m \tag{ 14-17 } $$ The 90 angles between meridians and parallels are preserved, but because the scale along the lines of longitude does not match the scale along the lines of latitude, the final projection is not conformal. Poles: Normally circular arcs, enclosing the same angle as the Frequently used in the ellipsoidal form for maps of the United States in the National Atlas of the United States, for thematic maps, and for world atlases. If $(\lambda - \lambda_0)$ exceeds the range $\pm180$, 360 should be added or subtracted to place it within the range. Krasovskiy (projection l) in 1922. Latitude lines are unequally The scale along the meridians is just the opposite, and in fact the scale factor along meridians is the reciprocal of the scale factor along parallels, to maintain equal area. In order to preserve area, the scale factor of a meridian at any given point is is a conic, equal-area projection, in which parallels are unequally spaced arcs with the same subscripts 1, 2, or none applied to $m$ and $\phi$ in equation (14-15), and 0, 1, 2, or none applied to q and $\phi$ in equation (3-12), as required by equations (14-12), (14-12a), (14-13), (14-14), and (14-17). The angles between the meridians are Choose a web site to get translated content where available and see local events and offers. origin of the rectangular map coordinates. Web browsers do not support MATLAB commands. software usage. The "one-sixth rule" places the first standard parallel at one-sixth the range above the southern boundary and the second standard parallel minus one-sixth the range below the northern limit. Suggested parallels for maps See Ellipsoids for more information, or execute In the latter case, both parallels are south of the islands, but they were chosen to include maps of the more southerly Canal Zone and especially the Philippine Islands. Parallels: Unequally spaced concentric circular arcs centered Then, It differs from the Lambert $$ Spacing of parallels decreases away from Conic Projection: Lambert, Albers and Polyconic - GIS Geography The pole is not the center of the circles, but is normally an arc itself. The cone of projection thereby becomes a plane. They may be on opposite sides of, but not equidistant from, the Equator. $\phi_1,\phi_2=$ standard parallels. Along meridians, scale follows an opposite pattern. Both poles are represented as arcs rather than as single points. You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. This is true of any regular conic projection. the central latitudes. might suppose, the Albers Equal Area Conic projection is a conic projection Other MathWorks country sites are not optimized for visits from your location. are chosen, not symmetric about the Equator, then a Lambert Equal-Area (14-12a); respectively. If equation (4-5) is used, $k$ will be found to be the reciprocal of $h$, satisfying the requirement for an equal-area projection when meridians and parallels intersect at right angles. Vitkovskiy (projection 11) in 1907, N.Ya. the scale factor of a meridian at any given point is the reciprocal See Ellipsoid size parameters for more information. 8 and 18N. Although neither shape nor linear scale is truly correct, the distortion of these properties is minimized in the region between the standard parallels. Used for the conterminous United States, normally using 2930' and 4530' as the two standard parallels. Albers projection - Wikipedia For improved computational efficiency using the series, see p. 19. \tag{ 14-11 } $$ \frac 1{2e} \ln{\left( \frac{1-e\sin{\phi}}{1+e\sin{\phi}}\right)}\right] \tag{ 3-16 } $$ Description adapted from J.P. Snyder and P.M. Voxland, An Album of Map Projections, U.S. Geological Survey Professional Paper 1453. It may be seen from equation (14-7), and indeed from equations (4-4) and (4-5), that distortion is strictly a function of latitude, and not of longitude. The default convention is to interpret this value as decimal degrees. vanishes along the two standard parallels. Based on your location, we recommend that you select: . Behrmann or other cylindric equal-area projections, if the two standard parallels are symmetrically placed north and south of the equator. Mapping Toolbox uses a different implementation of the standard Albers equal-area $$ \rho_0 = a(C-n q_0)^{1/2}/n \tag{ 14-12a } $$ In both of these limiting cases, the pole is a point. $$ \rho = R (C-2n\sin\phi)^{1/2}/n \tag{ 14-3 } $$ $$ \theta = n(\lambda-\lambda_0) \tag{ 14-4 } $$ MathWorks is the leading developer of mathematical computing software for engineers and scientists. the projection to be at 125 oE/20 oN and 25 oN We desire a map Polar Lambert azimuthal equal-area projection, if a pole is made the single standard parallel. (14-11) also apply unchanged. $$ \beta=\arcsin(q/\{1-[(1-e^2)/2e]\ln[(1-e)/(1+e)]\}) \tag{ 14-21 } $$ $$ x=\rho\sin\theta \tag{ 14-1 } $$ Last updated on Mar 31, 2023. The cone of projection thereby becomes a cylinder. FIGURE 20. Along parallels, scale is reduced between the standard parallels and increased beyond them. 14. Albers Equal-Area Conic projection | Eu, Mircea or 1:200000 which means 1 inch on the map equals 200,000 inches Along two selected parallels, called standard parallels, the scale is held exact; along the other parallels the scale varies with latitude, but is constant along any given parallel. - Albers Equal-Area Conic projection, with standard parallels 20 and 60 N. This illustration includes all of North America to show the change in spacing of the parallels. and a Lambert Azimuthal Equal-Area projection results. &+ (761e^6/45360+\dots)\sin{6\beta}+\dots \end{align} scale that way. Meridians, on the other hand, are equally spaced radii about The meridians are therefore radii of the circular arcs. Best results for regions predominantly eastwest in orientation and located in the middle latitudes. but $q$ is still found from equation (14-19). which parallels are unequally spaced arcs of concentric circles, United States Government Printing Office: 1989. Copyright 2017-2023, The PyGMT Developers. First If a pole is selected as a single standard parallel, the cone is a plane, and a Lambert Equal Area Conic projection is the result. European maps, its biggest success has been for maps of North Americaspecifically, for maps for the Here, too, constants $n, C$, and $\rho_0$ need to be determined just once for the entire map. $$ \rho_0 = R(C-2n\sin\phi_0)^{1/2}/n \tag{ 14-3a } $$ conterminous, Both poles are See Projection Units for more information. The parallels are spaced to retain the condition of equal area. Coordinates from a spherical datum can be transformed into Albers equal-area conic projection coordinates with the following formulas, where is the radius, is the longitude, the reference longitude, the latitude, the reference latitude and and the standard parallels: where Lambert equal-area conic As a rule of thumb, these parallels can be placed at one-sixth and five-sixths of the range of latitudes, but there are more refined means of selection. displayed parallels. where $\beta$, the authalic latitude, adapting equations (3-11) and (3-12), is found thus: $$ y=\rho_0-\rho\cos\theta \tag{ 14-2 } $$ To is free of distortion along the standard parallels. on the point of convergence. \theta = \arctan[x/(\rho_0-y)] For small areas, the overall distortion is minimal. There are other possible approaches. Although neither shape nor linear scale is truly correct, the distortion of these properties is minimized in the region between the standard parallels. For example, the USGS uses this conic projection for maps showing the conterminous United States (48 states). This projection, developed by Albers in 1805, is predominantly Parallels: Unequally spaced concentric circular arcs centered The angles between them are less than the true angles. To define the projection in GMT you need to cone becomes a cylinder and a Lambert Equal-Area Cylindrical projection You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. This projection Shape along the standard parallels is accurate and minimally distorted in the region between the standard parallels and those regions just beyond. Albers Conic Equal Area PyGMT pole is selected as a single standard parallel, the cone is a plane For the scale factor, modifying (4-25): (14-13), and $$ \phi = \arcsin\{[C-(\rho n/R)^2]/(2n)\} \tag{ 14-8 } $$ All rights reserved. This projection was presented by Heinrich Christian Albers in 1805 and it is also known as a Conical Orthomorphic projection. As with the spherical case, $\rho$ and $n$ are negative, if the projection is centered in the Southern Hemisphere. To convert coordinates measured on an existing map, the user may choose any meridian for $\lambda_0$. pole is an arc. Built with Sphinx lines. Defaults to "GRS80". To map a given region, standard parallels should be selected to minimize variations in scale. Free of angular and scale distortion only along the one or two standard parallels. of that along the parallel to preserve equal-area. Latitude lines are unequally Total range in latitude from north to south should not exceed 3035. $$ \theta = n(\lambda-\lambda_0) \tag{14-4} $$ 1805. Actually, if $e = 0, q= 2\sin\phi$. It Behrmann or other cylindric equal-area projections, if the two standard parallels are symmetrically . Meridians, on the other hand, are equally spaced radii about a common iteration does not converge, but $\phi = \pm90$, taking the sign of $q$. meridian is trimmed. These parallels provide for a scale error slightly less than 1 per cent in the center of the map, wit.h a maximum of 1 per cent along the northern and southern borders (Deetz and Adams, 1934, p. 91).
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