Which cartesian coordinate system would you use to best maintain shape?

The choice of a cartesian coordinate system that best maintains shape is the conformal projection. Conformal projections preserve angles, ensuring that shapes of small regions on the Earth's surface are accurately represented. This characteristic is vital for various applications, such as navigation, engineering, and detailed mapping, where maintaining the true shape of features is crucial.

In a conformal projection, local shapes are preserved even though size and area may be distorted when viewed over larger areas. This means that if you were to take a small area, the shape of that area would look very similar to its actual shape on the globe, making it suitable for topographic maps and other uses where detail and accuracy in local layouts are indispensable.

In contrast, other options, like the Equidistant Conic and Albers Equal Area projections, focus on preserving distances and areas rather than shape. The Transverse Mercator projection is also conformal, but it is specifically designed to minimize distortion along a central meridian and is most effective for narrow regions. Thus, while it's important to understand these other projections and their uses, when it comes to the specific requirement of maintaining shape accurately, the conformal projection stands out as the most appropriate choice.