proper/best type for storing latitude and longitude

Question

In a system level programming language like C, C++ or D, what is the best type/encoding for storing latitude and longitude?

The options I see are:

• IEEE

Answer 1

Great question!

I know this question is 9 years old now, and I only know a part of the answer you were seeking, but I just came here having a similar question, and many things have changed since that question was asked, such as hardware and GPSes available. I work with this subject frequently in firmware dealing with different kinds of GPSes in different kinds of applications, and have lost count of the hours (and days) I have spent working out "the best design" for different applications that I have worked with or developed.

As always, different solutions are going to provide benefits and costs, and ultimately, a "best design" is always going to be a "best fit" of the benefits and costs against system requirements. Here are some things that I have to consider when I ask the same question:

CPU Time Cost

If CPU does not have a built-in floating-point co-processor (as is the case with many microcontrollers), then dealing with 'float', 'double', and 'long double' can be extremely costly. For example, with one 16-bit microcontroller I work with regularly, a multiplication using 'double' values costs 326 CPU clock cycles, and a division costs 1193 clock cycles. Very expensive!

Accuracy Trade-Off

At the equator, a 'float' (IEEE-754 32-bit floating point value), needing to represent a signed degree value, assuming 7 "clean" significant decimal digits able to be represented, the change of one least-significant decimal digit (e.g. from 179.9999 to 180.0000) is going to represent a distance of about 11.12 meters. This may or may not meet hard system accuracy requirements. Whereas a 'double' (with 15 "clean" significant decimal digits represented, thus a change from 179.999999999999 to 180.000000000000) represents about 0.00011 mm.

Input Accuracy Limitations

If you're dealing with input from a GPS, how many digits of real accuracy are you getting, and how many do you need to preserve?

Development Time Costs

An IEEE-754 64-bit double-precision value ('double') and 32-bit single-precision value ('float') are VERY convenient to deal with in the C language since math libraries for both come with virtually every C compiler, and are usually very reliable. If your CPU comes with a hardware floating-point processor, this is an easy choice.

RAM and Storage Costs

If you have to keep a large number of these values in RAM (or storage e.g. MYSQL), available RAM (and storage space) might have an impact on the workability of the solution.

Available Data vs Required Data

One example I'm dealing with at this writing (the reason I came here to this question) is that I am dealing with a u-blox M8 GPS which is able to give me binary GPS information (saving the CPU overhead of translating ASCII NMEA sentences). In this binary format (called "UBX Protocol") latitude and longitude are represented as signed 32-bit integers, which representation is able to represent accuracy (at the equator) of down to about 1.11 cm. For example, -105.0269805 degrees longitude is represented as -1050269805 (using all 32 bits) and one LSb change represents about 1.11 cm change in latitude anywhere, and 1.11 cm longitude at the equator (and less at higher latitudes, in proportion to the cosine of the latitude). The application this GPS is in does navigation tasks, which (already existing and well-tested code) requires 'double' data types. Unfortunately, converting this integer to an IEEE-754 64-bit 'double' cannot be easily done just by moving the base-2 bits of the integer into the internal representation bits of the 'double' since the decimal shift to be performed is a base-10 decimal shift. Were it a base-2 decimal shift instead, then the base-2 bits of the integer could be moved into the bit-fields of the 'double' with very little translation required. But alas, this is not the case with the signed integer I have. So it is going to cost me a multiplication on a CPU that doesn't have a hardware floating-point processor: 326 CPU clock cycles.

double   ldLatitude;
int32_t  li32LatFromGps;
ldLatitude = (double)li32LatFromGps * 0.0000001;

Note this multiplication was chosen over this:

ldLatitude = (double)li32LatFromGps / 10000000.0;

because 'double' multiplication is about 3.6X faster than 'double' division on the CPU that I'm dealing with. Such is life in the microcontroller world. :-)

What would have been BRILLIANT (and may be in the future if I can spare the time on weekends) is if the navigation tasks could be done directly with the 32-bit signed integer! Then no conversion would be needed.... But would it cost more to do the navigation tasks with such an integer? CPU costs, probably much more efficient. Development time costs? That's another question, especially with a well-tested system already in place, that uses IEEE-754 64-bit 'double' values! Plus there is already-existing software that provides map data (using 'double' degree values), which software would have to be converted to use the signed integer as well -- not an overnight task!

One VERY interesting option is to directly (without translation) represent intersections between approximations of "rectangles" (actually trapezoids, which become triangles at the poles) using the raw latitude/longitude integers. At the equator these rectangles would have dimensions of approximately 1.11 cm east-west by 1.11 cm north-south, whereas at a latitude of say London, England, the dimensions would be approximately 0.69 cm east-west by 1.11 cm north-south. That may or may not be easy to deal with, depending on what the application needs.

Anyway, I hope these thoughts and discussion help others who are looking at this topic for "the best design" for their system.

Kind regards, Vic

Answer 2

The easiest way is just to store it as a float/double in degrees. Positive for N and E, negative for S and W. Just remember that minutes and seconds are out of 60 (so 31 45'N is 31.75). Its easy to understand what the values are by looking at them and, where necessary, conversion to radians is trivial.

Calculations on latitudes and longitudes such as the Great Circle distance between two coordinates rely heavily on trigonometric functions, which typically use doubles. Any other format is going to rely on another implementation of sine, cosine, atan2 and square root, at a minimum. Arbitrary precision numbers (eg BigDecimal in Java) won't work for this. Something like the int where 2^32 is spread uniformly is going to have similar issues.

The point of uniformity has come up in several comments. On this I shall simply note that the Earth, with respect to longitude, isn't uniform. One arc-second longitude at the Arctic Circle is a shorter distance than at the Equator. Double precision floats give sub-millimetre precision anywhere on Earth. Is this not sufficient? If not, why not?

It'd also be worth noting what you want to do with that information as the types of calculations you require will have an impact on what storage format you use.

Answer 3

http://www.esri.com/news/arcuser/0400/wdside.html
At the equator, an arc-second of longitude approximately equals an arc-second of latitude, which is 1/60th of a nautical mile (or 101.27 feet or 30.87 meters).

32-bit float contains 23 explicit bits of data.
180 * 3600 requires log2(648000) = 19.305634287546711769425914064259 bits of data. Note that sign bit is stored separately and therefore we need to amount only for 180 degrees.
After subtracting from 23 the bits for log2(648000) we have remaining extra 3.694365712453288230574085935741 bits for sub-second data.
That is 2 ^ 3.694365712453288230574085935741 = 12.945382716049382716049382716053 parts per second.
Therefore a float data type can have 30.87 / 12.945382716049382716049382716053 ~= 2.38 meters precision at equator.

Answer 4

If by "storing" you mean "holding in memory", the real question is: what are you going to do with them?

I suspect that before these coordinates do anything interesting, they will have been funnelled as radians through the functions in math.h. Unless you plan on implementing quite a few transcendental functions that operate on Deg/Min/Secs packed into a bit field.

So why not keep things simple and just store them in IEEE-754 degrees or radians at the precision of your requirements?

Answer 5

Might the problems you mentioned with floating point values become an issue? If the answer is no, I'd suggest just using the radians value in double precision - you'll need it if you'll be doing trigonometric calculations anyway.

If there might be an issue with precision loss when using doubles or you won't be doing trigonometry, I'd suggest your solution of mapping to an integer range - this will give you the best resolution, can easily be converted to whatever display format you're locale will be using and - after choosing an appropriate 0-meridian - can be used to convert to floating point values of high precision.

PS: I've always wondered why there seems to be no one who uses geocentric spherical coordinates - they should be reasonably close to the geographical coordinates, and won't require all this fancy math on spheroids to do computations; for fun, I wanted to convert Gauss-Krüger-Koordinaten (which are in use by the German Katasteramt) to GPS coordinates - let me tell you, that was ugly: one uses the Bessel ellipsoid, the other WGS84, and the Gauss-Krüger mapping itself is pretty crazy on it's own...

Answer 6

0.3 inch resolution is getting down to the point where earthquakes over a few years make a difference. You may want to reconsider why you believe you need such fine resolution worldwide.

Some of the spreading centres in the Pacific Ocean change by as much as 15 cm/year.