# Defining a scale number from the Helmert transformation

I know this is more bit of a cartographic issue, but maybe somebody would be able to help. I have an old map georeferenced to the openstreetmap with 50 identical points. I want to get the scale of the old map.

I am using Helmert 4-parameter tranformation (similarity) with the equations:
X=ax-by+X0
Y=ay+bx+Y0
where x and y are the old map coordinates in meters (same as coordinates used from openstreetmap), X0 and Y0 are translations and then there is one scaling and rotation.

Scaling parameter m could be obtained as:
m=sqrt(a^2+b^2)
Rotation is:
alpha=arctan(b/a)

The resolution of the old map is 400 dpi (0.0635 mm per pixel).

I have all the parameters counted now with the least squares method.
Does anybody know how to get the scale of the old map from these parameters? I want the scale in a common form 1 : M (1 : 600 000).
EDIT: Further on I will use iteratively reweighted least squares and I will be omitting those identical points with the biggest spatial error (I am working with very old maps which have mistakes). I will do this to make the definition of scale more accurate. That's why I need to use the transformation parameters for the map scale definition and not only to compare the distances.
Thanks a lot.

• One of the properties of your georeferenced map should be cell size, or the ground distance length/width of a pixel in whatever CRS you're using. If you know the mapped size of one pixel, and you know the image of the orginal map was at 400 pixels per inch, then 400 * cell size = ground distance, so the original map scale is 1 (inch) : ground distance. – Chris W May 26 '14 at 21:34
• Welcome to GIS.SE. You have an answer from @ChrisW above and another from me below. Unless we misunderstand something, notice we both ignore your similarity transformation, as it seems unrelated to the actual (simple-seeming) question. If you need to clarify things, please do so via the edit button. – Martin F May 27 '14 at 2:54
• Thanks a lot to both of you. I'm sorry I should have been more specific before. I need to use the transformation parameters because I want to make the map scale the most accurate and I don't want to rely on the map drawing itself in only one particular area. (@martinf I made the edit to the question) – Radek May 27 '14 at 9:56
• I'm still finding it hard to see how the transformation scale parameter is going to help -- unless it is from map units/distance to ground units/distance and in which case it gives you your answer directly. What are your helmert parameters? (Again, editing the Q is best.) – Martin F May 27 '14 at 17:22
• Under that approach you are saying the original map has no scale or it varies. No straight formula will give a single scale because rather than a fixed ratio it is a continuous surface type of data - scale between any two given points will vary depending on sample location. You would have to take n samples and work some statistics to determine a mean/average/approximate scale. But it sounds to me like you're trying to calculate distortion, not scale. A map is either to scale or not - mapping errors are a whole other issue. – Chris W May 27 '14 at 17:45

As a unitless ratio,

``````map scale = map length / ground length
``````

(assuming lengths are in same units)

Expressed in the conventional form, `1: M`,

``````M = 1 / map scale
= ground length / map length
``````

So, calculate the ground distance between two of your control points, measure the distance between the same two points on the old map, and (assuming the distances are in same units) divide one by the other. If your map measurement is in different units (not meters), first convert it to meters.

Note that this classic notion of map scale only makes sense if you have a printed or displayed map upon which you can measure distances. If this (digital) map gives you equivalent ground distances directly, then there is no ratio scale.

If, as you say, you are estimating, by least squares, the transformation scale between two coordinate systems (or two maps), then this represents, not the map-to-ground scale, but the system-to-system scale (or map-to-map distortion), as Chris says. A different kind of scale.

If you are trying to estimate the accuracy of the old map, and if you can assume the OpenStreets map is accurate, then you can use the results of your least squares helmert transformation analysis:

Each of your control points will have residuals -- difference between before and after adjustment values -- within the old coordinate system. These give an indication of accuracy. As you state, some of those are very large and you might consider removing them. (Be careful though; using all reasonable data is more realistic than cherry-picking only some.) An overall measure of accuracy would then be the "root mean square" (RMS) value.

• I will be using RMS value surely too. But the map scale is one of my necessary output. However I am now maybe a bit more familiarized with the terms scale and distortion. So the distortion would be different when using the different coordinate system for the ground data? – Radek May 28 '14 at 7:48
• @Radek -- Your "nominal" map scale is whatever is stated on the map. If the map (or any map) covers a large area, however, then there is distortion, ie, the actual scale changes throughout the map. Such a topic is map projections. It seems that you really should be asking several separate questions. – Martin F May 28 '14 at 16:37
• I think I got this in mind. I was intentionally choosing maps of small regions to have the influence of distortion as small as it could be and I would be able to omit the influence of a map projection. The area is about 300km diameter. – Radek May 28 '14 at 20:35

You cannot determine a single map scale from transformation formulas alone, only through statistical analysis of the results of those transformations. The transformation will alter the scale (distance) between any given two control points on the map and that resulting value may vary from any other control point pair. By analyzing the values from many pairs you can derive a mean/average/etc. global approximate scale, but that is its own analysis separate from the transformation process.