Standard Points and Lines in Map Projections

In order to be able to read the information a map conveys, we must be familiar with the distribution and size of the inevitable distortions. Otherwise, our knowledge will be deficient or even wrong. The paper first defines the terms standard point and standard line. The standard point is the point where the inevitable distortions caused by mapping are equal to zero. This definition can be visually interpreted as a Tissot distortion ellipse that becomes a unit circle. After that, it is natural to say that the standard line is composed of standard points. A large number of examples show that the map projection does not have to have standard points at all, but that it can have one such point, two such points or a whole line of standard points. In the latter case, it can be parallels, meridians or lines approximately parallel to the image of the middle meridian, as is the case with the Gauss–Krüger or transverse Mercator projection. In this article, formulas are derived by which the reader can mathematically determine standard points or lines, if such exist. The derivation of new mathematical formulas in the paper can be helpful to cartographers who develop a mapping application and may need to select a map projection for their application. The map projection may not be common and therefore the details of the projection’s standard point(s) or line(s) are not well documented. These equations then could be used in writing the code that mathematically derives the location of a standard point(s) or line(s) for a map projection and reports that location to the developer or the end user.