Which cartesian coordinate system is best for preserving distance?

The best choice for preserving distance among the options is the Equidistant Conic projection. This projection is specifically designed to maintain accurate distances from one or two selected points to all other points on the map. It does this effectively for mid-latitude regions and is often utilized for mapping areas that are elongated in an east-west direction.

The Equidistant Conic projection achieves its goal by ensuring that distances measured on the map accurately reflect true distances on the ground along certain lines. This quality makes it especially useful for applications such as resource management, transportation planning, and in scenarios where measuring distance accurately is critical.

While the other projections mentioned have their strengths, they are not primarily designed for distance preservation. For instance, the Azimuthal projection maintains direction but does not necessarily preserve distance universally. The Albers Equal Area projection focuses on maintaining area, ensuring that land sizes are represented accurately, but it compromises on distance accuracy. The Transverse Mercator projection preserves distance along the central meridian and is often used for mapping smaller areas or for navigation, but it doesn't maintain distance globally across the entire map. Thus, in terms of preserving distance across a wider area, the Equidistant Conic projection is the most suitable choice.