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...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 2πof 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 theMercator 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.展开更多
This paper explains that the terms“horizontal and vertical scales”are not appropriate in map projections theory.Instead,the authors suggest using the term“scales in the direction of coordinate axes.”Since it is no...This paper explains that the terms“horizontal and vertical scales”are not appropriate in map projections theory.Instead,the authors suggest using the term“scales in the direction of coordinate axes.”Since it is not possible to read a local linear scale factor in the direction of a coordinate axis immediately from the definition of a local linear scale factor,this paper considers the derivation of new formulae that enable local linear scale factors in the direction of coordinate x and y axes to be calculated.The formula for computing the local linear scale factor in any direction defined by dx and dy is also derived.Furthermore,the position and magnitude of the extreme values of the local linear scale factor are considered and new formulas derived.展开更多
If geodetic coordinates from an ellipsoid are included in the equations of a projection for mapping a sphere instead of geographical coordinates,the result will be a projection of the ellipsoid into a plane.This will ...If geodetic coordinates from an ellipsoid are included in the equations of a projection for mapping a sphere instead of geographical coordinates,the result will be a projection of the ellipsoid into a plane.This will slightly change the distortion distribution of the initial map projection.The question is to what extent the replacement of geographical with geodetic coordinates will affect this change.In this paper,we deal with conformal,equal-area and equidistant projections of the sphere,which we modify by using geodetic coordinates instead of geographical ones.The result will be an approximately conformal,approximately equal-area and approximately equidistant projection.It is shown that in this case the maximum distortion of the angles in approximately conformal projections will be approximately 23.09′,the maximum distortion of the area in approximately equal-area projections less than 0.7% and the maximum distortion of lengths in approximately equidistant projections less than 0.7%,therefore on the maps imperceptible.展开更多
文摘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 2πof 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 theMercator 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.
文摘This paper explains that the terms“horizontal and vertical scales”are not appropriate in map projections theory.Instead,the authors suggest using the term“scales in the direction of coordinate axes.”Since it is not possible to read a local linear scale factor in the direction of a coordinate axis immediately from the definition of a local linear scale factor,this paper considers the derivation of new formulae that enable local linear scale factors in the direction of coordinate x and y axes to be calculated.The formula for computing the local linear scale factor in any direction defined by dx and dy is also derived.Furthermore,the position and magnitude of the extreme values of the local linear scale factor are considered and new formulas derived.
文摘If geodetic coordinates from an ellipsoid are included in the equations of a projection for mapping a sphere instead of geographical coordinates,the result will be a projection of the ellipsoid into a plane.This will slightly change the distortion distribution of the initial map projection.The question is to what extent the replacement of geographical with geodetic coordinates will affect this change.In this paper,we deal with conformal,equal-area and equidistant projections of the sphere,which we modify by using geodetic coordinates instead of geographical ones.The result will be an approximately conformal,approximately equal-area and approximately equidistant projection.It is shown that in this case the maximum distortion of the angles in approximately conformal projections will be approximately 23.09′,the maximum distortion of the area in approximately equal-area projections less than 0.7% and the maximum distortion of lengths in approximately equidistant projections less than 0.7%,therefore on the maps imperceptible.
