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From circle to ellipse: How do Tisos show local deformation of maps?
In cartography, Tissot's indicatrix is a mathematical tool first proposed by French mathematician Nicolas-Auguste Tissot in 1859 and 1871. This concept is mainly used to describe local deformation caused by map projection. The core of the Tisos indicator is to project a circle of infinitesimal radius from a curved geometric model (such as the earth) and then observe the changes that occur on the map.
"Tissot proved that a circle after projection is no longer a circle, but is transformed into an ellipse."
Why do we need to use the Tissot index to analyze map deformation? Because deformation is inevitable on a map, the Tissot indicator shows how this deformation varies in different areas. Typically, a reticle indicator is drawn at each displayed intersection of longitude and latitude lines to facilitate observation of the local deformation of the map. These diagrams not only encourage us to think about the accuracy of the map, but also provide a basis for calculations to accurately represent the degree of deformation at each point.
The development and application of Tissot theory
Tissot's theory was developed in the context of map analysis, where usually geometric models represented the Earth, in the form of a sphere or ellipse. The Tisuo indicator can effectively display the linear, angular and area deformation of the map. Among them, linear deformation refers to the change in the length of an infinitely short line on the earth model when it is projected onto the map; if the ratio of its length deviates from 1, it can be determined that deformation exists.
"Different map projections have unique ways of preserving angles and areas, resulting in different shapes and orientations of their respective indexes."
The Tissot index can not only describe linear deformation, but also show area and angle deformation under different projections. In conservative-angle projections (such as conformal projections), the indices at each point are circular, and their size varies with geographic location. In a conservative area projection (such as the equal-area projection), all the indices have the same area, but their shape and orientation still vary with position.
Calculation and mathematical background of the extraction index
The calculation of the Tissot index is based on the theory of differential geometry, focusing on the three-dimensional coordinates of points on the Earth's surface. In practice, commonly used parameters such as scale factor and angular distortion vary with the projection method. These data are directly related to the deformation caused by the projection. By calculating the loss rate, researchers can obtain the specific change from a circle to an ellipse at any point on the map.
"A well-computed extraction metric is crucial because it helps us understand the destructive and local deformations exhibited by the map."
For non-conservative projections, the changes in curvature are no longer fixed; however, these changes open up entirely new directions in cartographic science research. The Tissot index describes this change, and each of its ellipses contains a deep understanding of the characteristics of map projections. Are there other graphics or concepts that can help us better understand the phenomenon of map deformation?
