How to solve overdetermined vector equation?

Hi,

I have an equation like this:

sin(a1)*sin(a2)=v1, -cos(a1)*sin(a2)=v2, cos(a2)=v3

v1, v2 and v3 are components of a vector I know. How can I calculate the equation to get the results of a1 and a2?

Thanks!

Answer 1

[Torsten]

Use lsqnonlin to minimize the function

F(a1,a2) = (sin(a1)*sin(a2)-v1)^2 + (cos(a1)*sin(a2)+v2)^2 + (cos(a2)-v3)^2

Best wishes

Torsten.

Answer 2

[Bjorn Gustavsson]

Use pen and paper, draw the standard little sketch for spherical-and-Cartesian coordinates...

Then you get:

a2 = acos(v3);
% sin(a1)/cos(a1) = -v1/v2, so
a1 = atan2(v1,-v2); % check the order of the arguments to atan2
% perhaps this is a numerically better estimate of a2:
a2 = atan((v1^2+v2^2)^(1/2)/v3);

HTH

[Torsten]

I get

a2 = -atan((v1^2+v2^2)^(1/2)/v3);

Best wishes

Torsten.

[Bjorn Gustavsson]

Realy? Oh, well - no reason for us to get all agitated about sign conventions for rotational angles...

(I've seen 4 different definitions of azimuth used in astronomy! Clock-wise from North, counter-clock-wise from East, clock-wise from East, clock-wise from East)