Discuss with respect to the theory and general appreciation of cartographical projections Lambert's orthomorphic and

Of course. Let’s delve into a detailed discussion of the Lambert Conformal Conic projection, often known by its older name, Lambert’s orthomorphic projection.

We will discuss it from two perspectives:

1. The Theory: The geometric and mathematical principles behind its construction and properties.
2. The General Appreciation: Its practical strengths, weaknesses, applications, and why it holds such an esteemed place in cartography.

A Note on Terminology: Orthomorphic vs. Conformal

First, it’s essential to clarify the term “orthomorphic.” Coined from Greek (orthos = right, morphe = shape), it literally means “right-shaping.” In modern cartography, the term “conformal” is used almost exclusively.

A conformal (or orthomorphic) projection is one that preserves angles locally. This means that if two lines on the globe intersect at a 90-degree angle, their corresponding lines on the map will also intersect at a 90-degree angle. A direct and highly valuable consequence of this property is that the shapes of very small features are preserved. A small island, for instance, will have the correct shape on the map, even if its size is distorted.

This preservation of shape comes at a cost: area is not preserved. On a conformal map, you cannot use the map to accurately compare the sizes of different regions.

1. The Theory of the Lambert Conformal Conic (LCC)

The LCC was designed by the brilliant Swiss mathematician and cartographer Johann Heinrich Lambert in 1772. It is a conic projection, meaning its construction is based on the idea of projecting the globe onto a cone.

The Geometric Concept: A Cone on a Globe

Imagine the Earth as a transparent globe with a light source at its center.

1. A cone is placed over the globe, like a dunce cap. The axis of the cone is aligned with the Earth’s axis of rotation (the North-South pole line).
2. The features of the globe (continents, latitude/longitude lines) are “projected” from the central light source onto the surface of the cone.
3. Finally, the cone is “unrolled” by cutting it along a line (a meridian), laying it flat to create the map.

This simple concept gives rise to the projection’s fundamental characteristics:

• Parallels of latitude are projected as concentric circles (or arcs of circles) on the cone.
• Meridians of longitude are projected as straight lines radiating from a single point—the apex of the cone.

Tangent vs. Secant Case (The Key to its Usefulness)

There are two ways the cone can be placed, and this is crucial to understanding the LCC’s genius.

• Tangent Cone: The cone touches the globe along a single line of latitude. This line is called the Standard Parallel. Along this line, the map scale is perfectly true, and there is zero distortion. Distortion increases as you move away (north or south) from this standard parallel.

• Secant Cone (More Common): The cone slices through the globe, intersecting it at two lines of latitude. These are the two Standard Parallels.

  • Along these two parallels, the scale is perfectly true and distortion is zero.
  • Between the two standard parallels, the scale is slightly smaller than reality (the map is compressed).
  • Outside the two standard parallels, the scale becomes progressively larger than reality (the map is stretched).

The secant case is far more useful because it distributes the unavoidable distortion over a much larger area. By carefully choosing the two standard parallels, a cartographer can create a map with extremely low distortion across a wide band of latitude.