如何将球面速度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面速度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
我有一个高度、经度、高度的速度向量，我想将它转换为笛卡尔坐标，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
OK 基于你之前的问题和长评论flow 让我们假设您的输入是:
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]
并想要输出:
x,y,z // Cartesian position [m/s]
vx,vy,vz // Cartesian velocity [m/s]
并且将有效转换到笛卡尔坐标以供您使用，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，您可以尝试推导出 lihe @MvG 建议的公式，但从您的新手错误来看，我强烈怀疑它会导致成功的结果.相反，您可以这样做:
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
lon,lat,alt方向向量(x,y,z)
obtain
lon,lat,altdirection vectors for your position(x,y,z)
这很容易，只需在 WGS84 位置使用一些小步增量转换为笛卡尔减法并归一化为单位向量.让我们称这些方向基向量为 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;
希望它足够清楚.像往常一样注意单位 deg/rad 和 m/ft/km，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.
顺便说一句 U,V,W 基向量形成 NEH 参考系，并且在相同的时间是MvG提到的方向导数.
Btw U,V,W basis vectors form NEH reference frame and in the same time are the direction derivates MvG is mentioning.
[Edit1] 更精确的转换
//---------------------------------------------------------------------------
//--- 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，它们需要基本的 3D 矢量数学，请参见此处的方程式:
However they require basic 3D vector math see here for equations:
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