Python——球面两点距离及两条直线夹角的计算

问题描述：
平常为了得出地理位置上两点的实际距离（譬如北京与杭州之间的实际距离），除了利用经纬度计算出两点的空间距离，还需要考虑地形因素。由于之间考虑地形造成误差较大，因此采用微分的办法来解决，简单来说就是将两点细分为多点间的距离（当然这个多点是有限的）。

图中计算AB之间的距离，可以计算出中间多个点位的距离（如AC），然后计算在AB直线上的投影。在此之前，需要计算出球面上两条直线间的夹角以及两点在球面上的距离。

1）球面上两条直线间的夹角
方法一：简易版，将球面弧线看成是直角坐标系下的直线，采用向量乘积。角度a = arc cos (a1a2/|a1||a2|)
方法二：将球面坐标系转为三维笛卡尔坐标系，然后利用向量乘积求角度。
这里考虑到向量的方向性及准确性，采用方法二会更好点。

def latlong_to_3d(latr,lonr):
    """Convert a point given latitude and longitude in radians to
    3-dimensional space,assuming a sphere radius of one."""
    return np.array((
        math.cos(latr) * math.cos(lonr),math.cos(latr) * math.sin(lonr),math.sin(latr)
    ))
def angle_between_vectors_degrees(u,v):
    """Return the angle between two vectors in any dimension space,in degrees."""
    return np.degrees(
        math.acos(np.dot(u,v) / (np.linalg.norm(u) * np.linalg.norm(v))))
# Convert the points to numpy latitude/longitude radians space
a = np.radians(np.array(A))
b = np.radians(np.array(B))
c = np.radians(np.array(C))

# The points in 3D space
a3 = latlong_to_3d(*a)
b3 = latlong_to_3d(*b)
c3 = latlong_to_3d(*c)
# Vectors in 3D space
a3vec = a3 - b3
c3vec = c3 - b3
# Find the angle between the vectors in 2D space
angle3deg = angle_between_vectors_degrees(a3vec,c3vec)

2)两点间的距离
S=R·arc cos[cosβ1cosβ2cos（α1-α2）+sinβ1sinβ2]
具体原理见球面两点距离原理

或是采用google map办法：
地球上任两点，其经度分别为A1、A2（E正，W负），纬度分别为B1、B2（N正，S负)。
令A0=（A1-A2）÷2，B0=（BI-B2）÷2
f=√sinB0×sinB0+cosB1×cosB2×sinA0×sinA0
S = 2Rf

在依次计算逐个站点在直线上的投影，然后考虑两点间的高程数据，进行累加得到AB间实际距离。
附上代码如下：

import pandas as pd
from dbfread import DBF
import os
import numpy as np
from math import radians, cos, sin, asin, sqrt
import math

def geodistance(lng1,lat1,lng2,lat2):
    lng1, lat1, lng2, lat2 = map(radians, [float(lng1), float(lat1), float(lng2), float(lat2)]) # 经纬度转换成弧度
    dlon=lng2-lng1
    dlat=lat2-lat1
    a=sin(dlat/2)**2 + cos(lat1) * cos(lat2) * sin(dlon/2)**2
    distance=2*asin(sqrt(a))*6371*1000 # 地球平均半径，6371km
    #distance=round(distance/1000,3)
    return distance

def latlong_to_3d(latr,lonr):
    """Convert a point given latitude and longitude in radians to
    3-dimensional space,assuming a sphere radius of one."""
    return np.array((
        math.cos(latr) * math.cos(lonr),math.cos(latr) * math.sin(lonr),math.sin(latr)
    ))
def angle_between_vectors_degrees(u,v):
    """Return the angle between two vectors in any dimension space,in degrees."""
    return np.degrees(
        math.acos(np.dot(u,v) / (np.linalg.norm(u) * np.linalg.norm(v))))

path = r'D:\20210208'
filetype = '.dbf'#指定文件类型
total=[]
def get_filename(path,filetype):
    name =[]
    final_name = []
    for root,dirs,files in os.walk(path):
        for i in files:
            if filetype in i:
                name.append(i.replace(filetype,''))#生成不带‘.csv’后缀的文件名组成的列表
    final_name = [item +'.dbf' for item in name]#生成‘.csv’后缀的文件名组成的列表
    return final_name#输出由有‘.csv’后缀的文件名组成的列表

path_all = get_filename(path,filetype)
for j in path_all:
    table = DBF(path + '/' + j) #, encoding='GBK')
    df = pd.DataFrame(iter(table))
    total_distance = 0.0
    total_distance_plain = 0.0
    A = (df['POINT_Y'][0],df['POINT_X'][0])
    B = (df['POINT_Y'][len(df)-1],df['POINT_X'][len(df)-1])
    a = np.radians(np.array(A))
    b = np.radians(np.array(B))
    for n in range(len(df)-1):
        C = (df['POINT_Y'][n],df['POINT_X'][n])
        D = (df['POINT_Y'][n+1],df['POINT_X'][n+1])
        c = np.radians(np.array(C))
        d = np.radians(np.array(D))
        print(j,A,B,C,D)
        # The points in 3D space
        a3 = latlong_to_3d(*a)
        b3 = latlong_to_3d(*b)
        c3 = latlong_to_3d(*c)
        d3 = latlong_to_3d(*d)
        # Vectors in 3D space
        b3vec = b3 - a3
        d3vec = d3 - a3
        if D == B:
            angle3deg_a = 0
        else:
            # Find the angle between the vectors in 2D space
            angle3deg_a = angle_between_vectors_degrees(b3vec,d3vec)
        
        # Vectors in 3D space
        a3vec = a3 - d3
        c3vec = c3 - d3
        if C == A:
            angle3deg_d = 0
        else:
            # Find the angle between the vectors in 2D space
            angle3deg_d = angle_between_vectors_degrees(a3vec,c3vec)
        if angle3deg_d < 90:
            angle3deg = angle3deg_a + angle3deg_d
        else:
            angle3deg = angle3deg_a + 180 - angle3deg_d
        x = geodistance(df['POINT_X'][n],df['POINT_Y'][n],df['POINT_X'][n+1],df['POINT_Y'][n+1])
        h = df['GRID_CODE'][n]-df['GRID_CODE'][n+1]
        x_t = x*cos(angle3deg)
        l = (x_t*x_t+h*h)**0.5
        total_distance = total_distance+l#地表距离
        total_distance_plain = total_distance_plain+x#弧面距离
    total.append((j,total_distance)) 
print(total)