如何将球速坐标转换为笛卡尔坐标 [英] How to convert a spherical velocity coordinates into cartesian
本文介绍了如何将球速坐标转换为笛卡尔坐标的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
我有一个在海拔，经度，海拔上的速度矢量，我想将其转换为笛卡尔坐标，vx，vy，vz.格式来自WGS84标准.
I have a velocity vector in altitude, longitude, altitude, I would like to convert it to Cartesian coordinates, vx,vy,vz. The format is from WGS84 standard.
这是公式
//------------------------------------------------------------------------------
template <class T>
TVectorXYZ<T> WGS84::ToCartesian(T latitude, T longitude, T elevation)
//------------------------------------------------------------------------------
{
double sinlat, coslat;
double sinlon, coslon;
sincos_degree(latitude, sinlat, coslat);
sincos_degree(longitude, sinlon, coslon);
const double v = a / sqrt(1 - WGS84::ee * sinlat*sinlat);
TVectorXYZ<T> coord
(
static_cast<T>((v + elevation) * coslat * sinlon),
static_cast<T>(((1 - WGS84::ee) * v + elevation) * sinlat),
static_cast<T>((v + elevation) * coslat * coslon)
);
return coord;
}
推荐答案
根据您之前的问题和长注释进行确认流让您假设您的输入是:
OK based on your previous question and long comment flow lets assume your input is:
lon [rad], lat [rad], alt [m] // WGS84 position
vlon [m/s], vlat [m/s], alt [m/s] // speed in WGS84 lon,lat,alt directions but in [m/s]
并希望输出:
x,y,z // Cartesian position [m/s]
vx,vy,vz // Cartesian velocity [m/s]
并且具有有效的直角坐标转换为您可以使用的位置，这是我的:
And have valid transformation to Cartesian coordinates for positions at your disposal this is mine:
void WGS84toXYZ(double &x,double &y,double &z,double lon,double lat,double alt) // [rad,rad,m] -> [m,m,m]
{
const double _earth_a=6378137.00000; // [m] WGS84 equator radius
const double _earth_b=6356752.31414; // [m] WGS84 epolar radius
const double _earth_e=8.1819190842622e-2; // WGS84 eccentricity
const double _aa=_earth_a*_earth_a;
const double _ee=_earth_e*_earth_e;
double a,b,x,y,z,h,l,c,s;
a=lon;
b=lat;
h=alt;
c=cos(b);
s=sin(b);
// WGS84 from eccentricity
l=_earth_a/sqrt(1.0-(_ee*s*s));
x=(l+h)*c*cos(a);
y=(l+h)*c*sin(a);
z=(((1.0-_ee)*l)+h)*s;
}
以及将向量归一化为单位大小的例程:
And routine for normalize vector to unit size:
void normalize(double &x,double &y,double &z)
{
double l=sqrt(x*x+y*y+z*z);
if (l>1e-6) l=1.0/l;
x*=l; y*=l; z*=l;
}
是的，您可以尝试推导lihe @MvG建议的公式，但是从您的菜鸟错误中，我强烈怀疑这将导致成功的结果.相反，您可以执行以下操作:
Yes you can try to derive the formula lihe @MvG suggest but from your rookie mistakes I strongly doubt it would lead to successful result. Instead you can do this:
-
获取位置
(x,y,z)
obtain
lon,lat,altdirection vectors for your position(x,y,z)
这很容易，只需在 WGS84 位置使用一些小的步长增量即可转换为笛卡尔减法并归一化为单位向量.让我们将这些方向基础向量称为U,V,W.
that is easy just use some small step increment in WGS84 position convert to Cartesian substract and normalize to unit vectors. Let call these direction basis vectors U,V,W.
double Ux,Uy,Uz; // [m]
double Vx,Vy,Vz; // [m]
double Wx,Wy,Wz; // [m]
double da=1.567e-7; // [rad] angular step ~ 1.0 m in lon direction
double dl=1.0; // [m] altitide step 1.0 m
WGS84toXYZ( x, y, z,lon ,lat,alt ); // actual position
WGS84toXYZ(Ux,Uy,Uz,lon+da,lat,alt ); // lon direction Nort
WGS84toXYZ(Vx,Vy,Vz,lon,lat+da,alt ); // lat direction East
WGS84toXYZ(Wx,Wy,Wz,lon,lat ,alt+dl); // alt direction High/Up
Ux-=x; Uy-=y; Uz-=z;
Vx-=x; Vy-=y; Vz-=z;
Wx-=x; Wy-=y; Wz-=z;
normalize(Ux,Uy,Uz);
normalize(Vx,Vy,Vz);
normalize(Wx,Wy,Wz);
将速度从lon,lat,alt转换为vx,vy,vz
convert velocity from lon,lat,alt to vx,vy,vz
vx = vlon*Ux + vlat*Vx + valt*Wx;
vy = vlon*Uy + vlat*Vy + valt*Wy;
vz = vlon*Uz + vlat*Vz + valt*Wz;
希望这很清楚.像往常一样，要小心deg/rad和m/ft/km单位，因为单位很重要.
Hope it is clear enough. As usual be careful about the units deg/rad and m/ft/km because units matters a lot.
Btw U,V,W基矢量形成 NEH参考系，同时也是方向导引 MvG 正在提及.
Btw U,V,W basis vectors form NEH reference frame and in the same time are the direction derivates MvG is mentioning.
[Edit1]更精确的转化
//---------------------------------------------------------------------------
//--- WGS84 transformations ver: 1.00 ---------------------------------------
//---------------------------------------------------------------------------
#ifndef _WGS84_h
#define _WGS84_h
//---------------------------------------------------------------------------
// http://www.navipedia.net/index.php/Ellipsoidal_and_Cartesian_Coordinates_Conversion
//---------------------------------------------------------------------------
// WGS84(a,b,h) = (long,lat,alt) [rad,rad,m]
// XYZ(x,y,z) [m]
//---------------------------------------------------------------------------
const double _earth_a=6378137.00000; // [m] WGS84 equator radius
const double _earth_b=6356752.31414; // [m] WGS84 epolar radius
const double _earth_e=8.1819190842622e-2; // WGS84 eccentricity
//const double _earth_e=sqrt(1.0-((_earth_b/_earth_a)*(_earth_b/_earth_a)));
const double _earth_ee=_earth_e*_earth_e;
//---------------------------------------------------------------------------
const double kmh=1.0/3.6; // [km/h] -> [m/s]
//---------------------------------------------------------------------------
void XYZtoWGS84 (double *abh ,double *xyz ); // [m,m,m] -> [rad,rad,m]
void WGS84toXYZ (double *xyz ,double *abh ); // [rad,rad,m] -> [m,m,m]
void WGS84toXYZ_posvel(double *xyzpos,double *xyzvel,double *abhpos,double *abhvel); // [rad,rad,m],[m/s,m/s,m/s] -> [m,m,m],[m/s,m/s,m/s]
void WGS84toNEH (reper &neh ,double *abh ); // [rad,rad,m] -> NEH [m]
void WGS84_m2rad (double &da,double &db,double *abh); // [rad,rad,m] -> [rad],[rad] representing 1m angle step
void XYZ_interpolate (double *pt,double *p0,double *p1,double t); // [m,m,m] pt = p0 + (p1-p0)*t in ellipsoid space t = <0,1>
//---------------------------------------------------------------------------
void XYZtoWGS84(double *abh,double *xyz)
{
int i;
double a,b,h,l,n,db,s;
a=atanxy(xyz[0],xyz[1]);
l=sqrt((xyz[0]*xyz[0])+(xyz[1]*xyz[1]));
// estimate lat
b=atanxy((1.0-_earth_ee)*l,xyz[2]);
// iterate to improve accuracy
for (i=0;i<100;i++)
{
s=sin(b); db=b;
n=divide(_earth_a,sqrt(1.0-(_earth_ee*s*s)));
h=divide(l,cos(b))-n;
b=atanxy((1.0-divide(_earth_ee*n,n+h))*l,xyz[2]);
db=fabs(db-b);
if (db<1e-12) break;
}
if (b>0.5*pi) b-=pi2;
abh[0]=a;
abh[1]=b;
abh[2]=h;
}
//---------------------------------------------------------------------------
void WGS84toXYZ(double *xyz,double *abh)
{
double a,b,h,l,c,s;
a=abh[0];
b=abh[1];
h=abh[2];
c=cos(b);
s=sin(b);
// WGS84 from eccentricity
l=_earth_a/sqrt(1.0-(_earth_ee*s*s));
xyz[0]=(l+h)*c*cos(a);
xyz[1]=(l+h)*c*sin(a);
xyz[2]=(((1.0-_earth_ee)*l)+h)*s;
}
//---------------------------------------------------------------------------
void WGS84toNEH(reper &neh,double *abh)
{
double N[3],E[3],H[3]; // [m]
double p[3],xyzpos[3];
const double da=1.567e-7; // [rad] angular step ~ 1.0 m in lon direction
const double dl=1.0; // [m] altitide step 1.0 m
vector_copy(p,abh);
// actual position
WGS84toXYZ(xyzpos,abh);
// NEH
p[0]+=da; WGS84toXYZ(N,p); p[0]-=da;
p[1]+=da; WGS84toXYZ(E,p); p[1]-=da;
p[2]+=dl; WGS84toXYZ(H,p); p[2]-=dl;
vector_sub(N,N,xyzpos);
vector_sub(E,E,xyzpos);
vector_sub(H,H,xyzpos);
vector_one(N,N);
vector_one(E,E);
vector_one(H,H);
neh._rep=1;
neh._inv=0;
// axis X
neh.rep[ 0]=N[0];
neh.rep[ 1]=N[1];
neh.rep[ 2]=N[2];
// axis Y
neh.rep[ 4]=E[0];
neh.rep[ 5]=E[1];
neh.rep[ 6]=E[2];
// axis Z
neh.rep[ 8]=H[0];
neh.rep[ 9]=H[1];
neh.rep[10]=H[2];
// gpos
neh.rep[12]=xyzpos[0];
neh.rep[13]=xyzpos[1];
neh.rep[14]=xyzpos[2];
neh.orto(1);
}
//---------------------------------------------------------------------------
void WGS84toXYZ_posvel(double *xyzpos,double *xyzvel,double *abhpos,double *abhvel)
{
reper neh;
WGS84toNEH(neh,abhpos);
neh.gpos_get(xyzpos);
neh.l2g_dir(xyzvel,abhvel);
}
//---------------------------------------------------------------------------
void WGS84_m2rad(double &da,double &db,double *abh)
{
// WGS84 from eccentricity
double p[3],rr;
WGS84toXYZ(p,abh);
rr=(p[0]*p[0])+(p[1]*p[1]);
da=divide(1.0,sqrt(rr));
rr+=p[2]*p[2];
db=divide(1.0,sqrt(rr));
}
//---------------------------------------------------------------------------
void XYZ_interpolate(double *pt,double *p0,double *p1,double t)
{
const double mz=_earth_a/_earth_b;
const double _mz=_earth_b/_earth_a;
double p[3],r,r0,r1;
// compute spherical radiuses of input points
r0=sqrt((p0[0]*p0[0])+(p0[1]*p0[1])+(p0[2]*p0[2]*mz*mz));
r1=sqrt((p1[0]*p1[0])+(p1[1]*p1[1])+(p1[2]*p1[2]*mz*mz));
// linear interpolation
r = r0 +(r1 -r0 )*t;
p[0]= p0[0]+(p1[0]-p0[0])*t;
p[1]= p0[1]+(p1[1]-p0[1])*t;
p[2]=(p0[2]+(p1[2]-p0[2])*t)*mz;
// correct radius and rescale back
r/=sqrt((p[0]*p[0])+(p[1]*p[1])+(p[2]*p[2]));
pt[0]=p[0]*r;
pt[1]=p[1]*r;
pt[2]=p[2]*r*_mz;
}
//---------------------------------------------------------------------------
#endif
//---------------------------------------------------------------------------
但是，他们需要基本的3D矢量数学公式，请参见此处的方程式:
However they require basic 3D vector math see here for equations:
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