The short communication discusses the interrelationships of loxodromes, isometric latitudes and the normal aspect of Mercator projection. It is shown that by applying the isometric latitude, a very simple equation of the loxodrome on the sphere is reached. The consequence of this is that the isometric latitude can be defined using the generalized longitude, and not only using the latitude, as was common until now. Generalized longitude is the longitude defined for every real number; modulo 2π of generalized and usual longitude are congruent. Since the image of the loxodrome in the plane of the Mercator projection is a straight line, the isometric latitude can also be defined using this projection. Finally, a new definition of the Mercator projection is given, according to which it is a normal aspect cylindrical projection in which the images of the loxodromes on the sphere are straight lines in the plane of the projection that, together with the images of the meridians in the projection, form equal angles, as do the loxodromes with the meridians on the sphere. The short communication provides additional knowledge to all those who are interested in the theory of maps in navigation and have a piece of requisite mathematical knowledge, as well as an interest in map projections. It will be useful for teachers and students studying cartography and GIS, navigation or applied mathematics.

We encounter the loxodrome in mathematics, cartography and seafaring. According to Britannica et al. [

There is relatively detailed cartographic literature on loxodromes, isometric latitude and Mercator projection [

Loxodrome and Mercator projection are closely related to navigation [

Isometric latitude (see details in

In this short communication, we start with the derivation of the loxodrome equation on the sphere in the geographic parameterization. Then, instead of geographic latitude, we introduce isometric latitude as a parameter. This shows how to arrive at a very simple equation of the loxodrome. It is a linear relationship between the isometric latitude and geographic longitude, with the fact that the longitude should be taken in a generalized sense, i.e., from the interval

After that, we consider the normal aspect Mercator projection of the sphere in the usual way and using the isometric latitude. Then we derive the equation of the loxodrome image in that projection. This gives us the possibility of a new interpretation of isometric latitude using the Mercator projection. Finally, the idea to approach the Mercator projection in a new way is presented. We define it as a normal aspect cylindrical projection in which the images of loxodromes on the sphere are straight lines in the plane of the projection that make the same angles as the images of the meridians in the projection as loxodromes with the meridians on the sphere.

Let us recall that for

Defines a sphere with its center at the origin of the coordinate system and the radius

The coefficients of the first differential form of this mapping are

The differential expressions for any curve on the sphere are

where

Latitude | Longitude | Azimuth |
---|---|---|

If we accept the relations from

Let it be

Which after integration gives

and it is the equation of the loxodrome connecting the latitude

Loxodromes on a sphere are generally spiral curves that wrap around each pole an infinite number of times (

If we start with the differential

After integration we get

We note that according to

If we want the loxodrome to pass through the point with geographical coordinates

Finally, if we want the relationship between

If

The isometric latitude

The purpose of the isometric latitude is to give a parametrization, in which the Gaussian fundamental coefficients

with the assumption that we took for the integration constant the value that gives

Note that

These relations are easily derived from the definition of isometric latitude

Furthermore, the differential

After integration, we get the equation of the loxodrome on the sphere in the form

where

where

If we want the loxodrome to pass through the point with coordinates

Meridians and parallels are special cases of loxodromes. For meridians,

Indeed, if we take

The equation of the loxodrome on the sphere expressed using geographic coordinates is

For constant values of

The Mercator projection is a conformal cylindrical projection. This means that the basic equations of the normal aspect projection are

where

where

It follows according to

i.e.,

and from there

where

Let us note at the end that the equations of the Mercator projection

The equation of the loxodrome on the sphere is

From

and then from

Although the geometric interpretations of latitude and longitude and geocentric and reduced latitude are well known, a similar interpretation of isometric latitude is not easy to find. For example, Heck [

Now we will give a new definition of the isometric latitude

A common approach to deriving the equations of the normal aspect Mercator projection is to look for a cylindrical projection that satisfies the conformality condition (

Let us start from the equations of any normal aspect cylindrical projection

For

where

Therefore, the equations of the normal aspect cylindrical projection, which has the property that every loxodrome on the sphere that forms an angle

In addition to the usual condition in map projections that

where we recognize the equations of the normal aspect Mercator projection.

It is known that instead of geographic latitude, it is convenient to introduce isometric latitude as a parameter when it comes to the issue of preserving angles [

The normal aspect of the Mercator projection of the sphere can be defined in the usual way or using isometric latitude. We have shown that the introduction of the isometric latitude is very clever when deriving the equation of the loxodrome image in that projection. Furthermore, it enabled a new definition of isometric latitude using the normal aspect Mercator projection.

When Mercator made his map, he had in mind the rectilinearity of the loxodrome, not conformality. The Mercator projection is usually defined as a cylindrical conformal projection, and the novelty of this paper is that this is a consequence of the new definition. Namely, this projection can also be defined as a normal aspect cylindrical projection in which the images of the loxodromes from the sphere are straight lines in the plane of the projection that form the same angles as the images of the meridians in the projection as the loxodromes with the meridians on the sphere. Thus, the article, in a certain way, connects Mercator’s original idea with today’s usual approach to his projection, as a conformal cylindrical projection. In this way, we enrich the theory of map projections and expand the horizons of the user’s knowledge.

The author would like to thank the anonymous reviewers for their useful comments.

The author received no specific funding for this study.

All the work was done by Miljenko Lapaine.

Not applicable.

The author declares that he has no conflicts of interest to report regarding the present study.