# Explaining scale decrease inside parallels and outside increase for Lambert conformal conic projection?

For a Lambert conformal conic projection, when projecting the cone on the globe, two parallels are formed and the scale decreases between the two parallels and increases outside them.

Why does this phenomenon of decreasing scale occur inside the parallels and increasing scale occurs outside them?

• Welcome to gis.stackexchange! Please edit the title of your question to include enough information for future visitors to be able to find this thread when looking for the same problem. Commented Apr 20, 2016 at 19:18

Between the two standard paralllels, the projection surface (the cone) is below the ellipsoid surface, so features must be reduced in size to fit on the cone. Above and below the standard parallels, the conic surface is above the ellipsoid so features must be enlarged to fit on the cone.

One illustration is here.

• I had not critically evaluated such pictures before now. (Snyder doesn't include one.) Most of those I can find look bogus, because they do not show specifically how the sphere is projected onto the cone. How the scale varies is not in the least apparent from the formulas, either. In particular, it sure looks like features have (eventually) to be shrunk to fit near the apex of the cone! Here's a reasonably accurate picture: mathworld.wolfram.com/ConicProjection.html . It shows that there is a unique parallel away from which scale increases. Commented Apr 21, 2016 at 20:49
• (Bill, I'm feeling stupid today and may be misunderstanding your comment). Yes, that's a nice picture for a tangent case. The page does remark (as does Snyder in his overview page on conic projections) that a geometrically projected conic projection is not often used. Albers and Lambert conformal conics aren't strictly "geometrically" developable. Commented Apr 21, 2016 at 22:49
• The Lambert conformal conic projection is based on a cone that is tangent (not secant) to the ellipsoid, right? So unless I'm missing something, the cone would never be below the surface? Commented Jun 24, 2023 at 17:36
• @Chris-RegenerateResponse Perhaps originally it was a tangent cone, but most usages use a secant cone instead as it better reduces overall distortion. You'll see some definitions with a single latitude of center parameter (sometimes called a standard parallel) but there's also a scale factor < 1.0 which effectively turns it into a secant case. Commented Jun 25, 2023 at 15:53
• @mkennedy I see, thanks for the clarification. Do you know any resources that go into detail about how such a scale factor is applied? Commented Jun 25, 2023 at 16:37

Think of map projections as the best way possible to preserve:

1. Area

2. Shape

3. Distance

4. Direction

All of these cannot be preserved perfectly at the same time. So a map projection such as the Lambert Conformal Conic best preserves those 4 attributes in mid-latitude regions.

Regarding the two lines you are asking about (the parallels), these lines are the only place where the scale is 100% correct. Outside of these lines, the map and it's contents will become distorted (areal deformation). Areal deformation decreases inside the two lines, thus providing optimal projection for data in this region.

• Most implementations of the Lambert conformal conic projection allow standard parallels to be anywhere. You've given one specific case for the northern part of the Northern Hemisphere. Commented Apr 21, 2016 at 0:00
• This answer fails to explain the specific phenomenon called out in the question; namely, why does scale decrease between the parallels and increase outside them? Commented Apr 21, 2016 at 20:46