How to transform cartesian or polar data to latitude longitude ?

# sorry if my english is bad

Hello, like the title of this question, i need to compute a Cartesian Points or Polar Points to Geographic coordinates ( Latitude and Longitude ) if the ThetaZeroLocation is on top and the x,y,z axis is like this.

where the 0,0,0 coordinate is the center of a radar. the radar location : latitude = -6.1275 , longitude = 106.6537, with height = 3 m above the ground.

i have the real data location of a target on the radar in all 3 coordinates, cartesian, polar, and latitude longitude. I want to transform the cartesian or polar data to geographic coodinates.

Im making a function that have several code to compute this problem,

for case 1 and 2, i get the reference from this site [https://vvvv.org/blog/polar-spherical-and-geographic-coordinates#:~:text=In%20order%20to%20match%20the,a%20longitude%20of%200%C2%B0.]

for case 3 [https://stackoverflow.com/questions/5674149/3d-coordinates-on-a-sphere-to-latitude-and-longitude]

and for case 4 [https://www.researchgate.net/post/How-to-determine-the-latitude-and-longitude-of-a-target]

function Data = Cart2LatLong(latRad,longRad,initData,n)
%X Y Z in Nautycal Mile
for i = 1:size(initData,1)
X = initData{i,1}(:,2);
Y = initData{i,1}(:,3);
Z = initData{i,1}(:,6)/6076; % ft to KM to NM
Zm = initData{i,1}(:,6)*0.3048; %ft to meter
switch n
    case 1
        r = sqrt(X.^2 + Y.^2 + Z.^2);
        Data{i,1}(:,1) = asin(Z./r)+latRad;
        Data{i,1}(:,2) = atan2(Y, X)+longRad;
        Data{i,1}(:,3) = X;
        Data{i,1}(:,4) = Y;
        Data{i,1}(:,5) = Z;
        Data{i,1}(:,6) = Zm;
    case 2
        r = sqrt(X.^2 + Y.^2 + Z.^2);
        Data{i,1}(:,1) = asin(Z./r)+latRad;
        Data{i,1}(:,2) = atan2(X,-Y)+longRad;
        Data{i,1}(:,3) = X;
        Data{i,1}(:,4) = Y;
        Data{i,1}(:,5) = Z;
        Data{i,1}(:,6) = Zm;
    case 3
        r = sqrt(X.^2 + Y.^2 + Z.^2);
        Data{i,1}(:,1) = atan2(Z,r)+latRad;
        Data{i,1}(:,2) = atan2(X,-Y)+longRad;
        Data{i,1}(:,3) = X;
        Data{i,1}(:,4) = Y;
        Data{i,1}(:,5) = Z;
        Data{i,1}(:,6) = Zm;
    case 4
        R_Tgt = initData{i,1}(:,4); % Data in NM ( Nautycal Mile )
        T_Tgt = initData{i,1}(:,5); % Data in NM
        Z_Tgt = Z*1.15078; % NM to mile
        R1 = 3959 + 0.0057556603249355915708;
        T1 = longRad;
        Phi1 = 90-latRad;
        R2 = R_Tgt*1.15078; % NM to mile
        T2 = 90-T_Tgt; % Because the zero location is on top, so the theta 90-theta
        Phi2 = 90-Z_Tgt;
        %% Extract NaN data 
        [R1,T1,Phi1] = extractNan3D(R1,T1,Phi1);
        [R2,T2,Phi2] = extractNan3D(R2,T2,Phi2);
        %% Transform polar data to cartesian
        [x1,y1,z1]= pol2car(Phi1,T1,R1);
        [x2,y2,z2]= pol2car(Phi2,T2,R2);
        x = x1+x2; y = y1+y2; z = z1+z2; %final cartesian
        %final polar
        R     = sqrt(x.^2+y.^2+z.^2);
        Theta = atan2(y,x);
        Phi   = atan2(sqrt(x.^2+y.^2),z.^2);
        Lat = 90 - Phi;
        Long = Theta;
        Data{i,1}(:,1)= Lat+latRad;
        Data{i,1}(:,2)= Long+longRad;
end
end
end
% coba function di matlab
function [x,y,z] = pol2car (phi,theta,rho)
x = rho.*sin(phi).*cos(theta);
y = rho.*sin(phi).*sin(theta);
z = rho.*cos(phi);
end
function [x,y,z] = extractNan3D(x1,y,z)
indnan = find(isnan(x1)); %find index that the data is NaN
x = x1(setdiff(1:length(x1),indnan));
y = y(setdiff(1:length(x1),indnan));
z = z(setdiff(1:length(x1),indnan));
end

so for example, the x,y,z,rho data in NM (Nautycal mile) and theta is

X = 0.578125000000000; Y = 11.0781250000000; Z = 7.55842659644503; rho = 11.0976562500000; theta = 2.99926757812500

the real latitude longitude is

latitude = -5.942989 ; longitude = 106.663383

but what i got from each program is

%n = 1 
latitude = -5.529427552;	longitude = 108.1723574;	
%n = 2 
latitude = -5.529427552;	longitude = 109.7431538;
%n = 3 
latitude = -5.614692417; 	longitude = 109.7431538;
%n = 4
latitude = 83.86984703; 	longitude = 106.4941534; %the longitude data result is the closest one to the real longitude

so am i have making a totally wrong code or just wrong in some line ? need help thank you.

[Zaidan Adenin Said]

this is the data that i use for this problem

% X (NM)             Y (NM)              Rho (NM)            The (deg)           Z (ft)       
0.578125000000000	11.0781250000000	11.0976562500000	2.99926757812500	45925
1.01562500000000	11.0546875000000	11.1015625000000	5.27893066406250	45925
1.46093750000000	11.0078125000000	11.1015625000000	7.55859375000000	45925
1.89843750000000	10.9375000000000	11.1054687500000	9.84375000000000	45925
2.32812500000000	10.8593750000000	11.1093750000000	12.1234130859375	45925
% Lat_Real    Long_Real
-5.942989	106.663383
-5.94338	106.67071
-5.94416	106.678168
-5.945331	106.685496
-5.946633	106.692692

Answer 1

[Zaidan Adenin Said]

So i think i found the answer. From the toolbox Automated driving ther is one function that work fine to me.

The function is [lat,lon,alt] = local2latlon(xEast,yNorth,zUp,origin)

so im just convert X,Y, and Z data with NM to meter. And because my data is already ENU form, so im not changing any direction. and the origin just a vector contain latitude, longitude, and the height of my radar.