I have two points in the sky and I have their coordinates in Azimuth and Elevation angles with respect to position O. I want to find the scalar angle between the two points w.r.t. position O. Thank you so much for your help!
[Math] Finding the angle between two points given their Azimuth and Elevation angles
angle
Related Solutions
Define the bearing angle $\theta$ from a point $A(a_1,a_2)$ to a point $B(b_1,b_2)$ as the angle measured in the clockwise direction from the north line with $A$ as the origin to the line segment $AB$.
Then,
$$ (b_1,b_2) = (a_1 + r\sin\theta, a_2 + r\cos\theta), $$
where $r$ is the length of the line segment $AB$. It follows that $\theta$ satisfies the equation
$$ \tan\theta = \frac{b_1 - a_1}{b_2 - a_2} $$
As suggested by @rogerl we can use the $\mathrm{atan2}$ function to compute $\theta$. Let
$$ \hat{\theta} = \mathrm{atan2}(b_1 - a_1, b_2 - a_2) \in (-\pi,\pi] $$
Then the bearing angle $\theta\in[0,2\pi)$ is given by
$$ \theta = \left\{ \begin{array}{ll} \hat{\theta}, & \hat{\theta} \geq 0\\ 2\pi + \hat{\theta}, & \hat{\theta} < 0 \end{array}\right. $$
Note that the equations are given in terms of Cartesian coordinates, so it is necessary to transform to screen coordinates. I believe the formula for $\hat{\theta}$ in terms of screen coordinates $(a_1,a_2)$ and $(b_1,b_2)$ is $\hat{\theta} = \mathrm{atan2}(b_1 - a_1,a_2 - b_2)$.
You could code this function in C++ as follows.
#include <cmath>
// Computes the bearing in degrees from the point A(a1,a2) to
// the point B(b1,b2). Note that A and B are given in terms of
// screen coordinates.
double bearing(double a1, double a2, double b1, double b2) {
static const double TWOPI = 6.2831853071795865;
static const double RAD2DEG = 57.2957795130823209;
// if (a1 = b1 and a2 = b2) throw an error
double theta = atan2(b1 - a1, a2 - b2);
if (theta < 0.0)
theta += TWOPI;
return RAD2DEG * theta;
}
Write down the actual coordinate vectors of the points, using your two angles. Take the dot product of those two vectors. Assuming we're on the unit sphere, this will give you $\cos\alpha$, where $\alpha$ is the angle between the two vectors. Taking $\arccos$ of this number will give you $\alpha$, and hence the length of the great circle arc joining the original two points.
Best Answer
The simplest is to convert from spherical to Cartesian coordinates and take the dot product (cosine of the angle).
https://en.wikipedia.org/wiki/Spherical_coordinate_system#Cartesian_coordinates