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The Miracle of Transformation on Maps: What is Tissot's Indicatrix?
In the field of cartography, Tissot's index was proposed by French mathematician Nicolas-Auguste Tissot in 1859 and 1871. This mathematical tool is designed to describe the local deformation caused by map projection. . Tissot showed that when a circle of infinitesimal radius is projected from a curved surface model (such as a globe) onto a flat map, the resulting geometry becomes an ellipse, with the axes of the ellipse representing the directions of maximum and minimum scaling of the point.
A Tisu indicator depicts the degree of deformation at a certain point, and often multiple Tisu indicators are placed throughout the map to show spatial changes in deformation.
Tiso indicators are not only used to show the deformation of different areas on the map, but are also the basis for accurate calculations. These calculations can more accurately represent the deformation size of each point. Because the mapped infinitesimal circles have the same area on the underlying surface model, the distortion imposed by the map projection is apparent through the Tiso index. Therefore, there is a one-to-one correspondence between the index and the metric tensor of the map projection coordinate transformation.
Tisot's theoretical development background was mainly concerned with the analysis of cartography, usually a geometric model representing the earth, in the form of a sphere or ellipsoid. The Tiso indicator shows the linear, angular and area deformation of the map:
The map distorts distances (linear deformation) in different directions, measured by the ratio of the length of an infinitesimal line segment on the projected surface to its length on the Earth model, a ratio called the scaling factor.
In terms of angle distortion, the angle on the map does not remain constant in the projection, which is expressed through the elliptical shape it produces.
The deformation of the map area is shown by the fact that the area on the earth model does not remain unchanged in the projection, which is also expressed by deforming the ellipse.
In a conformal map, the shape of the index is a circle, and changes in size depending on the location, and may also change in direction (according to the division of longitude and latitude). In the equal-area projection, the area of all Tiso indicators is the same, but the shape and direction change depending on the geographical location. In any projection, both area and shape will vary across locations along the map.
Although the mathematical analysis of TSO indicators has its complications, it can be understood as a tool to describe the deformation of the earth's surface. Through differential geometry, numerical calculation methods can be used to obtain the parameters of the index. Such calculation methods are becoming increasingly important among modern surveying and mapping workers, especially those professionals who need to calculate and analyze map projection performance.
With the rise of digital charting, the application of TSO indicators seems to be increasing. Whether for academic discussion or practical map preparation, this technology provides a precise and visual way to understand the deforming nature of maps. In the process of creating and using maps, can people become more sensitive to the impact of these deformations on our environment?
