将球体从coord1旋转到coord2，coord3会在哪里？ [英] Rotate a sphere from coord1 to coord2, where will coord3 be?

问题描述

我有三个坐标（lat，lon）在一个球体上。如果你想将整个球体从coord1旋转到coord2，那么coord3现在会被放置在哪里？

我一直在使用Great Circle（http：/ / /www.koders.com/python/fid0A930D7924AE856342437CA1F5A9A3EC0CAEACE2.aspx?s=coastline），但我创建了奇怪的结果，因为新计算的点在赤道一起组合在一起。这与我假设的方位角计算有关？

有人可能知道如何正确计算这个值吗？

编辑

我发现以下内容： http://www.uwgb.edu/dutchs/mathalgo/sphere0.htm

我想我现在需要计算笛卡尔坐标系（和0,0,0）中两点的旋转轴和旋转角度？我想这一定非常简单，与定义一架飞机并确定法线有关？有人可能知道我在哪里可以找到所需的方程式吗？

编辑2

Coord1和coord2是一个很棒的圈子。是否有一种简单的方法可以找到球体上大圆法线轴的位置？

编辑3

看起来我能解决它;）
http://articles.adsabs.harvard.edu//full/1953Metic...1...39L/0000039.000.html 做了诀窍。使用Visual Python我认为我现在已经解决了它：

<$ p $ <$ p

解决方案

p> ＃为了地理目的而首先描述的旋转：http://www.uwgb.edu/dutchs/mathalgo/sphere0.htm
＃http://stackoverflow.com/questions/6802577/ python-rotation-of-3d-vector
＃http://vpython.org/
from visual import *
from math import *
import sys

def ll2cart（lon，lat）：
＃http://rbrundritt.wordpress.com/2008/10/14/conversion-between-spherical-and-cartesian-coordinates-systems/
x = cos（lat）* cos（lon）
y = cos（lat）* sin（lon ）
z = sin（lat）
return x，y，z

def cart2ll（x，y，z）：
＃http：//rbrundritt.wordpress .com / 2008/10/14 / conversion-between-spherical-and-cartesian-coordinates-systems /
r = sqrt（（x ** 2）+（y ** 2）+（z ** 2） ）
lat = asin（z / r）
lon = atan2（y，x）
return lon，lat

def distance（lon1，lat1，lon2， lat2）：
＃http://code.activestate.com/recipes/576779-calculating-distance-between-two-geographic-points/
＃http://en.wikipedia.org/wiki / Haversine_formula
dlat = lat2 - lat1
dlon = lon2 - lon1
q = sin（dlat / 2）** 2 +（cos（lat1）* cos（lat2）*（sin（dlon / 2）** 2））
return 2 * atan2（sqrt（q），sqrt（1-q））

如果len（sys.argv）== 1：
sys.exit（）
else：
csv = sys.argv [1]

＃定义旋转的点A和B：
LonA =弧度（ （float（sys.argv [4]））
float LatB = r adians（float（sys.argv [5]））

＃A和B都是矢量
＃交叉积AxB是旋转极点矢量P：
Ax，Ay， Az = ll2cart（LonA，LatA）
Bx，By，Bz = ll2cart（LonB，LatB）
A =矢量（Ax，Ay，Az）
B =矢量（Bx，By， Bz）
P = cross（A，B）
Px，Py，Pz = P
LonP，LatP = cart2ll（Px，Py，Pz）

＃以弧度表示的旋转角度：
＃http://code.activestate.com/recipes/576779-calculating-distance-between-two-geographic-points/
＃http：//en.wikipedia。 org / wiki / Haversine_formula
RotAngle =距离（LonA，LatA，LonB，LatB）

f = open（csv，r）
o = open（csv [： - 4 （$）
（lon，lat）= line.strip（） .split（，）

＃将被转换的点C：
LonC =弧度（float（lon））
LatC =弧度（float（lat））

＃笛卡尔坐标点C：
Cx，Cy，Cz = ll2cart（LonC，LatC）
C =矢量（Cx，Cy，Cz）

＃C旋转到D：
D =旋转（C，RotAngle，P）
Dx，Dy，Dz = D
LonD，LatD = cart2ll（Dx，Dy，Dz）

o.write（str（degrees（LonD））+，+ str（degrees（LatD））+\\\
）


I have three coordinates (lat,lon) on a sphere. If you would rotate the whole sphere from coord1 to coord2, where will coord3 now be located?

I've been trying this out in Python using Great Circle (http://www.koders.com/python/fid0A930D7924AE856342437CA1F5A9A3EC0CAEACE2.aspx?s=coastline) but I create strange results as the newly calculated points all group together at the equator. That must have something to do with the azimuth calculation I assume?

Does anyone maybe know how to calculate this correctly?

Thanks in advance!

EDIT

I found the following: http://www.uwgb.edu/dutchs/mathalgo/sphere0.htm

I guess I now need to calculate the rotation axis and the rotation angle from the two points in cartesian coords (and 0,0,0)? I guess this must be very simple, something to do with defining a plane and determining the normal line? Does someone maybe know where I can find the needed equations?

EDIT 2

Coord1 and coord2 make a great circle. Is there an easy way to find the location of the great circle normal axis on the sphere?

EDIT 3

Looks like I was able to solve it ;) http://articles.adsabs.harvard.edu//full/1953Metic...1...39L/0000039.000.html did the trick.

解决方案

Using Visual Python I think I now have solved it:

# Rotation first described for geo purposes: http://www.uwgb.edu/dutchs/mathalgo/sphere0.htm
# http://stackoverflow.com/questions/6802577/python-rotation-of-3d-vector
# http://vpython.org/
from visual import *
from math import *
import sys

def ll2cart(lon,lat):
    # http://rbrundritt.wordpress.com/2008/10/14/conversion-between-spherical-and-cartesian-coordinates-systems/
    x = cos(lat) * cos(lon)
    y = cos(lat) * sin(lon)
    z = sin(lat)
    return x,y,z

def cart2ll(x,y,z):
    # http://rbrundritt.wordpress.com/2008/10/14/conversion-between-spherical-and-cartesian-coordinates-systems/
    r = sqrt((x**2) + (y**2) + (z**2))
    lat = asin(z/r)
    lon = atan2(y, x)
    return lon, lat

def distance(lon1, lat1, lon2, lat2):
    # http://code.activestate.com/recipes/576779-calculating-distance-between-two-geographic-points/
    # http://en.wikipedia.org/wiki/Haversine_formula
    dlat = lat2 - lat1
    dlon = lon2 - lon1
    q = sin(dlat/2)**2 + (cos(lat1) * cos(lat2) * (sin(dlon/2)**2))
    return 2 * atan2(sqrt(q), sqrt(1-q))

if len(sys.argv) == 1:
    sys.exit()
else:
    csv = sys.argv[1]

    # Points A and B defining the rotation:
    LonA = radians(float(sys.argv[2]))
    LatA = radians(float(sys.argv[3]))
    LonB = radians(float(sys.argv[4]))
    LatB = radians(float(sys.argv[5]))

# A and B are both vectors
# The crossproduct AxB is the rotation pole vector P:
Ax, Ay, Az = ll2cart(LonA, LatA)
Bx, By, Bz = ll2cart(LonB, LatB)
A = vector(Ax,Ay,Az)
B = vector(Bx,By,Bz)
P = cross(A,B)
Px,Py,Pz = P
LonP, LatP = cart2ll(Px,Py,Pz)

# The Rotation Angle in radians:
# http://code.activestate.com/recipes/576779-calculating-distance-between-two-geographic-points/
# http://en.wikipedia.org/wiki/Haversine_formula
RotAngle = distance(LonA,LatA,LonB,LatB)

f = open(csv,"r")
o = open(csv[:-4] + "_translated.csv","w")
o.write(f.readline())
for line in f:
    (lon, lat) = line.strip().split(",")

    # Point C which will be translated:
    LonC = radians(float(lon))
    LatC = radians(float(lat))

    # Point C in Cartesian coordinates:
    Cx,Cy,Cz = ll2cart(LonC,LatC)
    C = vector(Cx,Cy,Cz)

    # C rotated to D:
    D = rotate(C,RotAngle,P)
    Dx,Dy,Dz = D
    LonD,LatD = cart2ll(Dx,Dy,Dz)

    o.write(str(degrees(LonD)) + "," + str(degrees(LatD)) + "\n")

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