calculation the angle betwen X-axis created due coordinate system rotation - after projection the coordinate system on the plane

I have a coordinate system XYZ (black on figure) which can be rotated around

Z axis (Zrot angle) or around Y axis (Yrot angle) or around both axisies (around Z first and then around rotated (in first rotation around Z axis) Y axis).

E.g. after rotation only around Z axis we get X'Y'Z coordinate system (see fig).

I have also the direction (point P) defined by azimuth and evation (e.g. AZ, EL) in XYZ coordinate system (blue on figure).

Rotation code are below:

% geodetic elevation and azimuth in deg(El, AZ)

Esvo = 50;

Asvo = 30;

% MatLab elevation and azimuth

Esv = Esvo;

Asv = Asvo;

if Asvo < 180; Asv=-Asvo;

else Asv = 360-Asvo;

end

r=1

dtr = pi/180;

Es=Esv*dtr % deg to rad conversion

As=Asv*dtr % deg to rad conversion

% initial position in MatLab

[x0,y0,z0] = sph2cart(As,Es,r)

poz0 = [x0 y0 z0]

% coordinate system rotation angles in deg (Zrot=Aro1; Yrot=Ero1)

Ero1 = 10;

Aro1 = 40;

% AZ change after Zrot rotation (geodetic)

Asvn = Asvo-Aro1

% chenged AZ in MatLab

if Asvn < 180; Asvn=-Asvn;

else Asvn = 360-Asvn;

end

Esv=Esv*dtr % deg to rad conversion

Asvn=Asvn*dtr % deg to rad conversion

% position after rotation around Z-axis

[x1,y1,z1] = sph2cart(Asvn,Esv,r)

poz1 = [x1 y1 z1]

% rotation around Y-axis

ROTY = [ cosd(Ero1) 0 sind(Ero1);

0 1 0;

-sind(Ero1) 0 cosd(Ero1)];

% position after two rotation (in MatLab)

poz2 = poz1*ROTY;

x2 = poz2(1);

y2 = poz2(2);

z2 = poz2(3);

I would like to create a PLANE (red on figure) which is perpendicular to direction P (direction P will be normal to PLANE) and then calculate angle B

which is the difference betwen X and X' axisies orthogonally projected on PLANE (red on fig).

Any suggestions how can be it done?

Answer 1

[darova]

Here is an idea of how to project axis onto plane:

• Unit vector (normal) of a plane:
• Distance to plane:
• Axis projected onto plane:

Having both projected axes: