Efficient 3D Vector Reflection: Simplifying Unknowns in Mirror Systems

[jeffto]

Hello All,

Let say I have a system of two planar mirrors (M1 and M2), with a known input (I), and a desired output (O2). Here are the parameters:

Azimuth of input: A
Polar of input: B

Azimuth of normal of M1: A+180
Polar of normal of M1: X1 (unknown)

Azimuth of reflected ray from M1: A+180
Polar of reflected ray from M1: X2 (unknown)

Azimuth of normal of M2: X3 (unknown)
Polar of normal of M2: C

Azimuth of reflected ray from M2: D
Polar of reflected ray from M2: E

Where A, B, C, D, and E are known and dependent on the situation.
X1 would be halfway between B and X2 (mirror reflection geometry).

I have tried (still trying) to solve it by using the vector reflection formula and breaking everything down to Cartesian coordinates. The result equations are huge and hard to solve. Lots of trig is involved and it seems like I can't really isolate the unknowns for a few equations.

Before I delve into this method further, I was hoping someone could give me advice on a better method to solve for the unknowns.

Thanks for reading

 

[anathema]

!
Thank you for sharing your question with the community. I understand your frustration with trying to solve complex equations and isolating unknown variables. I would suggest approaching this problem using a ray tracing method.

First, we can simplify the problem by considering only one mirror at a time. Starting with M1, we can trace the path of the input ray (I) as it reflects off of M1 using the law of reflection. This will give us the direction of the reflected ray (X2) and the normal vector of M1 (A+180, X1).

Next, we can use the direction of the reflected ray (X2) and the normal vector of M1 (A+180, X1) to trace the path of the reflected ray as it hits M2. Again, we can use the law of reflection to find the direction of the ray after it reflects off of M2 (D, E).

Using this method, we can find the unknowns X1, X2, and X3 by solving for them in each step of the ray tracing process. This approach may be simpler and more intuitive than breaking everything down into Cartesian coordinates.

I hope this helps and good luck with your calculations! If you have any further questions or need more clarification, please don't hesitate to ask.