Projections of the Earth onto a Plane


The 'Map Projection Catalog' (see link below) is presenting a collection of 324 Map Projection Graphics !
A few of them are implemented in my Java applet.
The algorithms to compute bearings, distance, midpoint, and destination are from Movable Type Ltd (see link below).

Select a map projection frpm the menu:

Compute distance and bearings between to points given by latitude and longitude:

compute distance and

Select Latitude 1, Longitude 1 of point 1 (point), and Latitude 2, Longitude 2 of point 2 (point).
The distance (kilometers) is computed along a great-circle between the two points.
The Midpoint is the center of the two points on the great-circle.
Max. Lat. is the latitude of the closest point of the complete great circle to the north pole.
The loxodromic path is drawn by 9 intermediate point.

Compute destination point given distance and bearing from start point:

destination and bearing

Bearings are in the range 0° to 360°, clockwise from North (90°=E, 180°=S, 270°=W).
The Earth's radius is assumed to be 6371 km.

The value of "Scale" is the ratio of the great circe distance of the two points (km) and the length of the black line connecting the points on the plane of projection (pixels).

Compute loxodrome given bearing from start point:


Select Latitude 1, Longitude 1, and the bearing of the loxodrome. The path is computed and marked every 100 km (larger mark every 500 km), for a total length of 40,000 km.

For some maps a standard parallel can be chosen, where the projection shows no distortion:


Numerical output:

map angle Map Angle is the azimuth (bearing) from North of the line connecting start and destination point on the map.
The orthodromic distance or great-circle distance is the shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the destination point does not remain constant.
A loxodrome or rhumb line is an arc crossing all meridians of longitude at the same angle, i.e. a path with constant bearing as measured relative to true or magnetic north.
Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles.
linear scale map earth
Scale P1 --> P2 is the ratio of the orthodromic distance and the distance D on the map.
The ratios N-S and E-W are referring to ±5° from the intersection point of the central meridian and the equator.
                                          map earth    
AM0 is the area on the map (in pix^2) of a 10°-square around the intersection point of the central meridian and the equator.

AE0 is the area on the Earth (in km^2) of a 10°-square around the intersection point of the central meridian and the equator.

AM0/AE0 is the ratio in units of pix2/km2.
area map earth

area map Earth
The green area (e.g. 10° by 10°) is selected by the latitude and longitude menues.

The area AE on the Earth is computed by the length of the corresponding meridian (1111.9 km per 10° of latitude), and the mean length of the limiting parallels.

The area AM on the map is approximated by the quadrilateral, and computed by the coordinates:
A=0.5*|[(x3-x1)(y4-y2) +(x4-x2)(y1-y3)]|.

You can check the equal-area property of a map projection by comparing the AM/AE quantity of 10°x10° quadrilaterals.

Loxodrome Orthodrome length

1. Stabius-Werner

The originators were Johannes Werner (1466–1528), a parish priest in Nuremberg, refined and promoted this projection that had been developed earlier by Johannes Stabius (Stab) of Vienna around 1500.

Distances are correct from one pole, as well as along all parallels.
Area preserving (Equal-Area).
Parallels are concentric circles about the North Pole.
No distortion along the central meridian.

rho= 90° - latitude
E = longitude*cos(latitude)/rho
x = r*sin(E)
y = 90 - r*rho*cos(E)

2. Mercator

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator (Geert de Kremer) in 1569.

Lines of constant course, known as rhumb lines or loxodromes, are represented as straight segments that conserve the angles with the meridians.
Conformal: preserving the angles and the shapes of small objects.
Distorts the size of objects as the latitude increases from the Equator.

K = π/180
x = K*longitude
y = 0.5*r*log[(1+sin(K*latitude))/(1-sin(K*latitude))]

3. Schmid Net

The projection can only represent one hemisphere of the earth.

Area preserving (Equal-Area)
angle distorting: most of the grid lines do not intersect at right angles.

px = cos(latitude)*sin(longitude/2)
pz = cos(latitude)*cos(longitude/2)
x  = 2*r*px/sqrt(1+pz)

py = sin(latitude)
pz = cos(latitude)*cos(longitude/2)
y = r*py/sqrt(1+pz)

4. Hammer-Aitow

This projection was described by Ernst von Hammer in 1892. It displays the world within a 2:1 elliptical outer shape, and is used even today for world maps.

2:1 elliptical outer shape

The vertical coordinates (longitude) are divided by two, and the values of the meridians from the center are doubled:

px = cos(latitude)*sin(longitude/2)
pz = cos(latitude)*cos(longitude/2)
x  = 2*r*px/sqrt(1+pz)

py = sin(latitude)
pz = cos(latitude)*cos(longitude/2)
y  = r*py/sqrt(1+pz)

5. Stereographic

x = r*cos(latitude)*cos(longitude)
y =  r*sin(latitude)

6. Winkel's Tripel

Winkel's Tripel projection is one of three projections proposed by Oswald Winkel in 1921. The projection is It is the arithmetic mean of the equirectangular projection and the Aitoff projection:
Winkel's Tripel alhorithm averages x and y-coordinates of the equidistant cylindrical projection, this time with the Aitoff projection's.

Not equal-area, not conformal.
Parallels are straight at Equator and poles, curved elsewhere;
scales are constant, but not equal, only at the Equator and central meridian.

K = π/180
alpha = arccos(cos(latitude)*cos(0.5*longitude))
si = sin(alpha)/(alpha/K)
With the conventional standard parallel: phi1=±2/π
With the equator as the standard parallel: phi1=0;
x = R*0.5*[longitude*cos(phi1) + 2*cos(latitude)*sin(0.5*latitude)/si]
y = R*0.5*[latitude + sin(latitude)]/si)

7. Mollweide

The projection was first published by mathematician and astronomer Karl Brandan Mollweide (1774 – 1825) of Leipzig in 1805.

2:1 elliptical outer shape
accuracy of proportions in area

Rx = 2*sqrt(2)/π
Ry = sqrt(2)
K = π/180
x =Rx*lambda*K*cos(t)
y =Ry*sin(t)
t is given by the equation: 2*t + sin(2*t) = π*sin(latitude),
which has to be solved by an iteration:
tn+1 = tn - [2*tn + sin(2*tn) - π*sin(latitude)] / [2 + 2*cos(2*tn*k))]

8. Littrow

This projection was developed by Joseph Johann von Littrow in 1833.
It is the only conformal retroazimuthal map projection. As a retroazimuthal projection, the Littrow shows directions, or azimuths, correctly from any point to the center of the map.

x = R*sin(longitude)/cos(latitude)
y = R*cos(longitude)*tan(latitude)

9. Van der Grinten

The Van-der-Grinten projection was published in 1904 by Alphons J. van der Grinten (1852–1921).
It projects the entire Earth into a circle

not equal-area
not conformal
polar regions are subject to extreme distortion

k = π/180

A = 0.5*abs(π/longitude - longitude/π)
theta = arcsin(2*latitude/π);       
G = cos(theta)/(sin(theta)+cos(theta)-1)
P = G*(2/sin(theta)-1)
N = P*P+A*A
Q = A*A+G
x = R*π*(A*(G-P*P) + sqrt(A*A*(G-P*P)*(G-P*P)-N*(G*G-P*P)))/N
y = R*π*(P*Q - A*sqrt((A*A+1.0)*N-Q*Q))/N

10. Lambert Equatorial Azimuthal Equal-Area

root = R*sqrt[2/(1 + cos(latitude)*cos(longitude))]
x = root*sin(longitude)*cos(latitude)
y = root*sin(latitude)

11. Lambert Equatorial Azimuthal Equidistant

alpha = arccos[cos(latitude)*cos(longitude)]
x = alpha*R*cos(latitude)*sin(longitude)/sin(alpha)
y = alpha*R*sin(latitude)/sin(alpha)

12. Braun

This stereographic cylindrical projection was designed by Carl Braun (1867).

asymmetric scaling horizontally and vertically

K = π/180
x = R*K*longitude
y = 2*R*tan(latitude/2)

13. Sinusoidal

The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection.


x = R*longitude*cos(latitude)
y = R*latitude

14. Eckert III

Max Eckert-Greifendorff presented six map projections in 1906. The latitudes are parallel lines in all six projections. In Eckert III meridians are elliptical, the outermost are semicircles, and the length of the central meridian is half the length of the equator.

not equal-area
not conformal

x = 0.5*R*longitude*[1 + sqrt(1-latitude2/902)]
y = R*latitude

15. Gnomonic

A points on the surface of the Earth's sphere is projected from the sphere's center to a point in a plane that is tangent.
Antipodal points appear at the same point in the plane. Thefore it can only be used to project one hemisphere.

x = R*tan(longitude)
y = R*tan(latitude)/cos(longitude)

16. Polyconic

K = π/180
E = longitude*sin(latitude)
x = R*sin(E)/tan(latitude)
y = R*[K*latitude + (1 - cos(E))/tan(latitude)]

17. Albers

This projection was published in 1805 by Heinrich C. Albers (1773–1833). The cone is intersecting the globe in two standard parallels (or is tangent in one standard parallel).

scale and shape are not preserved
distortion is minimal between the standard parallels.

Standard Parallels: 0°, 60°
n = 0.5*sin(60)
theta = n*longitude
rho = sqrt(1-2*n*sin(latitude))/n
rho0 = 1.0/n
x = R*rho*Math.sin(K*theta)
y = R*[rho0 - rho*cos(theta)]

18. Bonne

The projection is named after Rigobert Bonne (1727–1795), although it was in use prior to his birth.

Parallels of latitude are concentric circular arcs, and the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted.
Meridians: Central meridian is a straight line. Other meridians are complex curves connecting points equally spaced along each parallel of latitude and concave toward the central meridian.

rho = 1/tan(paral) + K*paral - K*latitude
E = K*longitude*cos(latitude)/rho
x = R*rho*sin(E)
y = R*[1/tan(paral) - rho*Math.cos(E)]

18. Lambert Conic

This projection is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication.

Straight line drawn on a Lambert conformal conic projection approximates a great-circle route between endpoints for typical flight distances.
Two parallels can be assigned unit scale, with scale decreasing between the two parallels and increasing outside them. This gives the map two standard parallels.

K = π/180
n = sin(paral)
F = cos(paral)*tann(π/4+0.5*K*paral)
rho0 = F/tann(π/4)
rho = F/tan(π/4+0.5*K*latitude)
x = R*rho*sin(n*longitude);
y = R*[rho0-rho*cos(n*longitude)]

19. Craster

This projection was developed by John Evelyn Edmund Craster in 1929; it was further developed by Charles H. Deetz and O.S. Adams in 1934. It was presented independently in 1934 by Putnins as his P4 projection.


meridians are equally spaced parabolas
parallels are unequally spaced straight parallel lines
free of distortion only at the two points where the central meridian intersects the 36º46' parallels

K = π/180
x =  R*sqrt(3/π)*K*longitude*[2*cos(2*latitude/3)-1]
y = R*sqrt(3*π)*sin(latitude/3)

Web Links
Map Projections (CSISS)

Map Projections (Wolfram MathWorld)

Stereographic Projection (Wolfram MathWorld)

Werner projection (Wikipedia)

Schmidt net (Wikipedia)

Hammer projection (Wikipedia)

Winkel tripel projection

Three Modifications for Azimuthal Projections

Calculate distance, bearing and more between Latitude/Longitude points (Movable Type Ltd)

Mollweide Projection (Wolfram MathWorld)

Mollweide projection (Wikipedia)

Rhumb line (Wikipedia)

Geocart Projections: Littrow

Van der Grinten projection (Wikipedia)

Online rhumb line calculations using the RhumbSolve utility

Two Aspects for Two Arbitrary Azimuthal Projections

2016 J. Giesen

updated: 2016, Aug 11