# [Proj] Pixel area errors

strebe at aol.com strebe at aol.com
Tue Nov 21 00:32:42 EST 2017

```Ken,

For any projection, at any point that is not a singularity, you compute the Jacobian matrix at that point. From the Jacobian, you can determine the scale factor in any direction. The Jacobian is a convenient way to compute the Tissot indicatrix, which is the measure of distortion at the point.

See, for example:

Piotr H. Laskowski (1989), “The Traditional and Modern Look at Tissot’s Indicatrix”,

The American Cartographer, 16:2, 123-133
http://dx.doi.org/10.1559/152304089783875497

If you only work in one projection, you can probably simplify quite a bit. For example, on spherical Mercator, the scale factor s is

s = sec φ

where φ is the latitude.

This is purely local distance I’m talking about here, which seems to be what you want.

Regards,
— daan Strebe

-----Original Message-----
From: Ken Mankoff <mankoff at gmail.com>
To: proj <proj at lists.maptools.org>
Sent: Mon, Nov 20, 2017 9:14 am
Subject: [Proj] Pixel area errors

Dear List,

I've just learned that GRASS has a function to provide the true scale for a given cell. From this, I can determine the map projection error.

My understanding is that for a given projection, the "lat_ts" term of the proj4 string describing the projection is where 1 m in GRASS is actually 1 m. Elsewhere, errors grow. I'd like to track these.

I asked the following on the GRASS list and was advised that this list might be a better place to ask:

Is the scale error only in longitude and never in latitude? Or can it be in both? Is there a way to find the error for a length error rather than an area error? Specifically, for a vector line with an arbitrary heading?

Thank you,

-k.
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