Final answer:
Distance and direction are quite accurate on conic projections, and small segments can approximate great circle routes. Such geometric approximations are used in various fields, including map projections like Mercator projections and astronomical measurements like parallax.
Step-by-step explanation:
Direction and distance are relatively accurate on conic projections, and straight lines drawn on them approximate great circle routes if distances are not great. This is because conic projections are designed to preserve angles, making them a good choice for mapping areas with a large east-west extent. When we consider small segments of such projections, the straight-line distance on the map can provide an excellent approximation of the distance over the earth's surface. This approximation is akin to how thin lens equations in physics give precise results, limited chiefly by the accuracy of the given information.
Similarly, Mercator projections are another type of map projection where latitude lines are stretched to match the length of the equator, which is useful for navigation since it preserves angles and makes straight lines on the map approximate rhumb lines, or lines of constant compass bearing.
In applications like astronomy and navigation, precise measurements and approximations are crucial. For example, the method of measuring parallax for determining distances in astronomy relies on the small angle approximation when the angle is small, which is similar to approximating a small arc as a straight line over short distances. The parallax is the angle subtended at an observation point by the distance between two points, and it's used to derive distances to celestial objects.
These methods and concepts in both geography and astronomy highlight the importance of mathematical precision and approximation in achieving accurate results, whether in measuring distances between points on a map or determining the vast distances between celestial objects.