Derivation of Cylindrical Projections

Next we show you exemplarily the proceeding of a mathematical construction of map projections for example the cylindrical projection. Thereby angles will be specified in radian measure. The globe will be "scaled down" to a unit sphere with radius 1. It will always be considered

     -   ≤  u  ≤      and    -  / 2  ≤  v  ≤   / 2 ,

thereby u will be interpreted as geographic longitude and v as geographic latitude.

1. Stage: equidistant cylindrical projection (Marinos from Tyre)

The most simple cylindrical projection is the so called Plate Carrée Map. The Plate Carrée Map is one of the oldest maps at all (about 100 A.D.). It arises from the simple correlation of geographic co-ordinates (u,v) of the unit sphere and the cartesian co-ordinates (x,y) of the plane. Thus we define

      x = u    and    y = v .               (1)

 equidistant cylindrical projection (Marinos from Tyre)

Clearly spoken: A tangent-(rotation-)cylinder is slipped over the (unit) sphere. The equator as a tangential circle is mapped length-preserving. The length of the meridians determines the height of the cylinder, i.e. there isn't any distortion of length (scale error) along the North-South-direction. The scale error along the West-East-direction upon the latitude v is easy to verify:

                Cosinus

The radius of the circle assigned to the latitude v is deluted to the radius of the equator. In case of the (unit) sphere with radius 1 it means that the radius of the parallel of latitude v ( in other words cos(v) ) is deluted to 1. Thus the dilation factor is  1 / cos(v) . Because the circumference is proportional to the radius of the circle we have found the scale error along the West-East-direction upon the latitude v  in  1 / cos(v). This distortion measure is valid for all cylindrical projections which are presented in this topic.

2. Stage: equal-area cylindrical projection (Archimedes)

We will get interesting variations of the Plate Carrée Map (1) if we introduce a non-linear scaling along the North-South-direction (y-direction):

      x = u    and    y = f(v)                (2)

with an adequate function f .

In case of  f(v) = sin(v)  we get the equal-area cylindrical projection which originates from Archimedes.

 equal-area cylinderical projection (Archimedes)

The area-fidelity of this cylindrical projection is a result of the Theorem of Archimedes which says that

a spherical cap (spherical segment) is equal-area to a cylinder shell with the same height and radius.

                Sinus

For this cylindrical projection the scale error along the West-East-direction upon the latitude v is given by  1 / cos(v)  analogue the Plate Carrée Map (1).

The scale error along the North-South-direction can not so easily be determined. Evidently the scaling (2) from above does not cause a constant distortion along the whole meridian. Therefore we look at the scale error of a meridian arc local between the latitudes v1 and v2, in other words we analyse the difference quotient

         difference quotient .

For small arcs    we get the locale scale error within the neighbourhood of a point of the latitude v1; so with    we get the derivation

     differential quotient

as the local North-South-distortion of a map (2) upon the latitude v. The differentiability is therefore a concrete postulation on the scaling f above. More reasonable postulations which can be constituted by geometry are:  f(0) = 0 (standardisation), f  is monotonic increasing, f  is odd (i.e. symmetry with  f(v) = - f(-v)), derivation of  f  is monotone.

Thus the equal-area cylindrical projection according to Archimedes has the local North-South-distortion  f ' (v) = cos(v)  within the neighbourhood of the latitude v.

3. Stage: conformal cylindrical projection (Mercator)

After we have learned something about the equidistant and the equal-area cylindrical projection we can now devote to the third possible differential-geometrical property of maps: the conformality of the cylindrical projection.

The conformal cylindrical projection is actually one of the most important (and historical most interesting) maps. It was designed as a sea chart by G. Mercator in the 16th century. Its great importance to seafaring is still unbroken.

 conformal cylinderical projection (Mercator)

The projection equations of the last-mentioned cylindrical projections are quasi self-explanatory. In the present case it is different: How has the scaling f  (2) to be constructed so that our cylindrical projection will be conformal.

At first we analyse in brief the term »conformal«. Certainly you remember the definition of the term »similar« associated with triangles in geometry lessons. Two triangles (in the plane) are called similar if the line proportion of related sides are equal. But this is true if and only if the accordant angles are equal.

Well, conformal projections are   l o c a l   affinity mappings, e.g. small figures onto the globe appear as similar figures onto the map (preserving shape); in particular, a small circle around any point onto the globe must be projected as a circle onto the map. Thus, the local North-South-distortion  f ' (v)  and the scale error along the West-East-direction upon the latitude v must be equal. Thus we get

        

as a necessary condition for the conformality of a cylindrical projection. This condition is also commensurate.

As a result of the solution of this integral we get for the scaling f:  f (v) = ln tan(  / 4  +  v / 2 ).  Therewith Mercator’s map is finally described.

Generalisation to secant-(rotation-)cylinder

In the conclusions above we assumed a tangent-(rotation-)cylinder. At this the  equator is exclusively mapped as a single length-preserving parallel of latitude (standard parallel or standard line).

Whereas in case of a secant-(rotation-)cylinder we have two standard lines  + v0 and - v0. The unwinding of the secant-(rotation-)cylinder causes a map which is upset along x-direction. For this kind of cylindrical projection it obtains:

        x = u * c        mit  0 < c  ≤  1 .               (3)

Completely analogue to the above considerations about the scale error along the West-East-direction upon the latitude v  we get:

        c = cos(v0)

as dilation factor along x-direction and for all three presented maps above cos(v0) / cos(v)  instead of  1 / cos(v)  as scale error along the West-East-direction upon the latitude v. Depending on the differential-geometrical property of the cylindrical projection this conclusion causes also a modification of the scalings  f (v)  along the North-South-direction.

1. For the equidistant cylinderical projection we get:  f (v) = v .
2. For the equal-area cylinderical projection we get:   f (v) = sin(v) / cos(v0) .
3. For the conformal cylinderical projection we get:   f (v) = cos(v0) * ln tan(  / 4  +  v / 2 ) .

Please check up these determinations on the basis of the above proceedings.

The usage of the (secant-)cylindrical projections is suitable in need of good values of distortion in the neighbourhood of the parallels of latitude  + v0  or  - v0   instead of the equator.

  
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