μ=ds'/ds
The length ratio is a variable, which varies not only with the position of the point, but also with the direction. Length ratio refers to the proportion of tiny line segments in a certain direction.
Usually, when studying the length ratio, we don't study the length ratios in all directions one by one, but only the length ratios in some specific directions, that is, the maximum length ratio (a) and the minimum length ratio (b), the warp length ratio (m) and the weft length ratio (n). If the longitude and latitude lines are orthogonal after projection, the length ratio of the longitude and latitude lines is the maximum and minimum length ratio. After the projection, the longitude and latitude lines are not orthogonal, and the included angle is θ, then the relationship between the longitude and latitude line length ratios M and N and the maximum and minimum length ratios A and B is as follows: According to Apollon's Nya theorem in analytic geometry,
m2+n2=a2+b2
m n sinθ=a b
The length deformation can be explained by the length ratio. The so-called length deformation is the difference between the length ratio (μ) and 1. The length deformation shown in Table V is: v=μ- 1.
Therefore, the length deformation can be divided into positive and negative, and the length deformation is positive, which means that the length increases after projection; Negative length deformation means that the length becomes shorter after projection; If the length deformation is zero, the length is not deformed. Area ratio: the ratio of the tiny area (deformed ellipse area) dF' on the projection plane to the corresponding tiny area (tiny circle area) dF on the spherical surface. Taking the area of deformed ellipse on the projection plane as dF'=abπ, and the area of tiny circle on the corresponding sphere as dF= 12π as an example, and taking p as the area ratio, then:
P=dF'/dF=abπ/π=ab
The above formula shows that the area ratio is equal to the product of the length ratio in the main direction. If longitude and latitude are the main directions:
P=mn
If the latitude and longitude direction is not the main direction, the area ratio
P=mnsinθ(θ is the included angle of latitude and longitude after projection)
The area ratio is a variable, which varies with the position of the point.
The area deformation is the difference between the area ratio and 1, which is expressed by Vp.
Vp=p- 1
There is positive and negative area deformation, and the area deformation is zero, indicating that there is no area deformation after projection, and the area deformation is positive, indicating that the area increases after projection; The area deformation is negative, indicating that the area decreases after projection. The scale marked on the map is called the main scale. When drawing the latitude and longitude network by map projection method, the ellipsoid of the earth should be reduced at a specified scale. For example, make a map of 1: 1 10,000, first shrink the earth by 1 10,000 times and then project it on the plane, then 1: 1 10,000 is. Due to the deformation after projection, main scale can only remain at the point or line that is not deformed after projection, and other places are either larger than main scale or smaller than main scale. So what is larger or smaller than the main scale is called local scale.
Pay attention to the difference between length scale, length deformation and map scale.
5. Equal deformation line
All kinds of projections have errors or distortions. And the deformation of different points is often different, so as to observe and understand the deformation distribution in the drawing area. Isomorphic lines are often used to represent the deformation distribution characteristics of drawing areas. Isomorphism line is the connecting line of points with equal deformation values, which is drawn in the grid of latitude and longitude lines according to the calculated values of various deformations (such as p, w), such as area and other deformation lines.
Isomorphic lines have different shapes on different projection diagrams. In azimuth projection, because there is no deformation in the projection center, the deformation increases gradually from the projection center, and the isomorphic lines are concentrically distributed. Equal deformation line pass