What is the mathematical tool used to characterize local distortions arising from map projections?

Tissot's Indicatrix is the mathematical tool specifically designed to illustrate and characterize the local distortions that occur due to map projections. This tool represents distortion through a small circle (indicatrix) at different points on a map, which is transformed into an ellipse when the projection is applied. The shape and orientation of this ellipse indicate how angles and areas are distorted by the projection at that point.

By analyzing the Tissot's Indicatrix, geodetic engineers can assess the accuracy of the map projection they are using and understand how distances, angles, and areas will be represented on the resultant map. This is crucial in geodesy and cartography for ensuring that spatial representations retain meaningful relationships to the real-world features they depict.

In contrast, the error ellipse characterizes the uncertainty of measurements rather than map projection distortions, the cartographic circle is not an established mathematical tool for evaluating distortions, and the metric tensor is a broader concept in differential geometry used to describe distances and angles in curved space, rather than specifically addressing map projection distortions.