Calculating Direction of Specular Reflected Sun Beam

[steil]

Homework Statement

Given the position of the sun (zenith and azimuth angle) and the orientation of a plane (inclination angle and orientation angle). What would be the correct equation to calculate the direction (zenith and azimuth angle) of the specular reflected sun beam as seen from that plane?
Example: Sun position: azimuth angle 160° (where azimuth is given clockwise from North (0°)), zenith angle 30°. Plane: Inclination angle 40°, orientation 180° (south).

Homework Equations

Law of reflection.

The Attempt at a Solution

I know that both the incoming and reflected beam form the same angle with respect to the normal to the plane and that the azimuthal directions of the incoming and reflected beam lie both on a plane (thus are shifted by 180°), but did not suceed to figure out the detailed equation to erform the above mentionned calculation for an abritrary sun and plane position. I know the law of reflection (http://en.wikipedia.org/wiki/Specular_reflection) but did not manage to calculate it right.
Please help! Thanks a lot in advance!

 

[KataKoniK]

Thank you for bringing up this interesting question. I would like to offer some insights on how to approach this problem.

Firstly, we need to understand the concept of the azimuth and zenith angles. The azimuth angle is the horizontal angle measured clockwise from North (0°) to the direction of the sun. The zenith angle, on the other hand, is the angle between the sun's position and the vertical direction.

Now, in order to calculate the direction of the reflected sun beam as seen from the plane, we can use the law of reflection. According to this law, the angle of incidence (i) is equal to the angle of reflection (r). In this case, the angle of incidence is the angle between the incoming sun beam and the normal to the plane, and the angle of reflection is the angle between the reflected sun beam and the normal to the plane.

To calculate the angle of incidence, we can use basic trigonometry. The incoming sun beam forms a right triangle with the normal to the plane and the line connecting the sun's position and the point where the beam hits the plane. The angle of incidence is then equal to the inverse tangent of the opposite side (sun's position) over the adjacent side (normal to the plane).

Once we have the angle of incidence, we can use the law of reflection to calculate the angle of reflection. This angle will be equal to the angle of incidence, but it will be on the opposite side of the normal to the plane. This means that we need to add 180° to the azimuth angle of the incoming sun beam to get the azimuth angle of the reflected sun beam.

To calculate the zenith angle of the reflected sun beam, we can use the law of reflection again. The zenith angle will be equal to the complement of the angle of incidence (90° - i).

In summary, the equation to calculate the direction of the reflected sun beam as seen from the plane would be:

Azimuth angle of reflected sun beam = Azimuth angle of incoming sun beam + 180°
Zenith angle of reflected sun beam = 90° - Inverse tangent (Opposite side/Adjacent side)

I hope this helps in your calculations. Let me know if you have any further questions.