June 28, 2022

More than the Coordinates

Written by: Sam Knight


Geographic Calculator, synonymous with coordinate transformations and conversions, seems to be all about the coordinates and the accuracy and precision of the positions. While this isn’t inaccurate, there are also other components of the calculations that are very useful but easily overlooked with the focus just on the positions. Some of these extra pieces are easily explored in Geographic Calculator’s Interactive Conversions and Point Database Conversions jobs. Particularly within projected coordinate systems, there are some calculations about distances and headings relating to North that become very important when taking measurements to the field. Likewise, when we start measuring distances on the surface of our coordinate systems at elevation, there are other scales that must be accounted for when translating distances to the physical world in which we operate.

Point Database Conversion job in Geographic Calculator
The Point Database Conversion job in Geographic Calculator generates scale, convergence, and magnetic declination values for each point in the target projected coordinate system.

Scaling distances

When converting or transforming data into a projected coordinate system, we often ignore the fact that there are native distortions involved in the coordinate grids. That is to say, since the earth is round and the projections are flat, something has to give, and one of the things that might be sacrificed is the direct accuracy of distances. We call the distortion around any point “Scale Factor” and sometimes referred to as “Grid-Scale Factor.” Scale factor changes throughout the mapped area, and how much it changes and in what directions that change takes place depends directly on the specific coordinate system. Because Scale factors only apply in projected coordinate systems, when calculating data out to a geodetic (Lat/Long) system in Geographic Calculator, you won’t see the option for Scale factor on the output. Scale is essentially a coefficient that is applied to measurements at a specific location for how those distances translate from the map to the real world.

Getting your bearings

Along with Scale, another output we find in projected coordinate systems is “Convergence,” an angular correction to headings measured from the projected map into real-world true North headings. There are multiple versions of “North” in a projected coordinate system, and Convergence relates to what is known as “Grid North.” Often, for those new to geodetics, this is confused with Magnetic North which is different. Grid North directly relates to how the projection shows the north pole versus where the true north actually is on the globe. A common assumption is that north on a map is “up,” which is true when you are talking about Grid North. True North may be in a specific place on the map, but Grid North is always at the top of the page. The angular relationship of a particular coordinate to those places on the map is the Convergence, and this is a critical concept for heading in the right direction in the field.

The other, other North

Fieldwork often relies on magnetic compass readings to properly orient directions. We just talked about how the map has its own version of North that differs from the North Pole, but unfortunately, neither of those are particularly useful for orienting yourself in the physical world.  The magnetic pole is in a significantly different location than True North, and to complicate it further; it actually moves through time. The difference between True North and Magnetic North, is referred to as “Magnetic Declination,” and it requires both a horizontal position and a date to properly calculate the relationship. Since magnetic declination requires time, by default it is turned off out of the box in Geographic Calculator and is easily toggled on in the Preferences.  Once enabled, you will see both Magnetic Declination as well as Convergence for any projected coordinate system outputs. Both are needed to make a measurement from the map and translate it to a proper heading in the real world using a compass or magnetometer.

Magnetic Declination in Geographic Calculator
Created as grids and displayed in Global Mapper, scale, convergence, and magnetic declination are shown for UTM Zone 19N. Using this visual the change variation in these values across the transverse Mercator zone can be seen.

Scaling new heights

Earlier, we talked about horizontal scales found in projections. Like the multiple versions of north, scaling also takes place in multiple ways within a map. As elevation increases or decreases away from the ellipsoid surface, we introduce “Orthometric Scale,” or “Height Scale.”  Remembering that we are mapping a curved surface, as we move off of that surface, those arcs we measure on the surface will stretch out as we go above, and shrink as we go below the ellipsoid. This is completely independent of the grid-scale and is not at all dependent on the projection. In fact, orthometric scale depends only on the relationship to the datum surface, so it affects geodetic (Lat/Long) coordinate systems exactly the same as a projected system. A very common calculation done to translate distances from the map to the real world involves what is casually known as “reducing to ground,” and this factors in both the horizontal grid-scale and the orthometric scale in what is called “Combined Factor.” Combined Factor is simply those two scales multiplied together, and each of those pieces is available for output on a calculation involving the relevant scales.

Putting it all together

When you look at a coordinate conversion as just the positions, you lose the context of the map, and makes it easy to forget the abstractions that are involved in every coordinate space. Taking a look at the specific distortions found between the references really helps give context for the coordinate space, and remember that we are not just talking about strictly cartesian grids in mapping and GIS. It also helps to reinforce the idea that maps are not necessarily “inaccurate” due to the distortions we see but rather that they are simply different and predictable mathematical models for identifying the relationships of objects, considering positions as well as the shapes, areas, and distances that define our world.

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