指向大圆段的距离

Question

指向大圆段的距离

geometrydistancepointwgs84great-circle

4

我希望能够计算从一个由经度和纬度给出的点到由两个点给出的线段（大圆的一部分）的距离。所有坐标都是以WGS84表示的。

我知道如何在笛卡尔坐标系中计算，但不知道如何在球面上计算。请问有人可以提供这个公式吗？

- user2033412

1

我认为你在mathematics.stackexchange上可能会更有好运。 - Timothy Groote

我投票关闭此问题，因为它与编程或软件开发无关，而是与[math.se]有关。 - Pang

距离d应该是穿过球体的直线，还是大圆弧线段？ - hypehuman

@hypehuman：它也应该是一个大圆段。 - user2033412

2个回答

2

这是横向距离，这里有详细描述。

dxt = asin( sin(δ13) ⋅ sin(θ13−θ12) ) ⋅ R
    where
 δ13 is (angular) distance from start point to third point
     θ13 is (initial) bearing from start point to third point
     θ12 is (initial) bearing from start point to end point
     R is the earth’s radius

您可以使用给定页面上的公式计算所需的距离和方位角。

distance
a = sin²(Δφ/2) + cos φ1 ⋅ cos φ2 ⋅ sin²(Δλ/2)
c = 2 ⋅ atan2( √a, √(1−a) )
d = R ⋅ c
where   
 φ is latitude, λ is longitude, R is earth’s radius (mean radius = 6,371km);

bearing
θ = atan2( sin Δλ ⋅ cos φ2 , cos φ1 ⋅ sin φ2 − sin φ1 ⋅ cos φ2 ⋅ cos Δλ )
    where
φ1,λ1 is the start point, φ2,λ2 the end point (Δλ is the difference in longitude)

请注意，将角度传递给三角函数时需要使用弧度制。

- MBo

非常抱歉，我会删除我的评论。 - Blindman67

5

为什么这被接受了？这是指一个点到大圆的距离，而不是 OP 所请求的大圆段。 - boycy

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- Spektre · Accepted Answer

spherical to 2D Cartesian

if the distance is not too far and not around pole singularities you can convert both line segment and line emitting from your point and perpendicular to your segment to spherical coordinates (if they are not already) and use the 2 angles as Cartesian space (ignoring radius).
1. compute intersection point
2. convert back to spherical
3. compute arclength between point and intersection
  
  hard to say if you're using sphere or WGS84 or what ....
Cartesian 3D

lets have arc segment AB, sphere of radius R and center C (ideally (0,0,0)) and point P then I see it like this:
1. find intersection point P' between plane ABC and its normal going through point P in 3D Cartesian
2. project it back on sphere surface
  
  For spherical surface is this easy as the projection means just to change the vector P'C length to R (if the sphere is centered around (0,0,0)).
```
P'' = (R*(P'-C)/|P'-C|) + C
```
3. compute arclength between the 2 points |P-P''|
  
  Also simple for spherical surface just compute the angle between vectors P-C and P''-C
```
ang = acos(dot(P-C,P''-C)/(R*R)); // [radians]
```
  and convert to arclength
```
d = ang*R; // [same Units as R]
```