Which of the following best represents the arc distances used in geographical calculations?

The choice of ellipsoidal distance is the correct representation of arc distances used in geographical calculations because it takes into account the shape of the Earth, which is not a perfect sphere but rather an oblate spheroid or ellipsoid. This means that when measuring distances over the Earth's surface, especially over large areas, the curvature of the Earth must be considered to obtain accurate results.

Ellipsoidal distances utilize mathematical models that describe the Earth's shape, allowing for precise calculations of distances along the surface that reflect the true path that would be traversed. This is particularly important in surveying and geodesy, where accurate positioning and distance measurements are crucial.

In contrast, spherical distance simplifies calculations by assuming the Earth is a perfect sphere, which can lead to inaccuracies. Plane distance assumes a flat surface, which is suitable for very small areas but not appropriate for larger distances due to the curvature of the Earth. Grid distance relates to measurements on a flat grid system, often used in mapping, which again does not account for the Earth's curvature over larger distances. Therefore, ellipsoidal distance is the most appropriate choice for accurate geographical calculations.