The Miracle

This is the last push.  Are you guys ready? If you answer is yes, you're a champ and if you're answer is no, let's just pretend you never said that.

As you remember, the goal of our research was to create a mirror that would image close to 180 degree field of view with minimum distortion.  So for, we have looked at single mirror, and various 2-mirror combinations, but we strongly suspect that the answer lies in a 3-mirror combination.  We learned that the rectifying mirror is great for reflecting the floor with minimum distortion. So the rectifying mirror is the permanent member of our mirror combination.  We also learned that the forward looking mirror did a great job for reflecting the ceiling.  So, only if we could rotate the forward looking mirror to reflect the  surrounding wall, instead of the ceiling, it would be superb. But, if you go back and read the section on the forward looking mirror again, you'll remember that the forward looking mirror only works when the light rays are parallel to its optical axis.  Therefore, we need a 45 degree linear mirror to not only change the path of light and hit the forward looking mirror, but to make sure that the light rays would hit the forward looking mirror parallel to the x-axis.

But these are all nice theories.  Do they really work as we want them in reality? Let's see!

We will first start with the rectifying mirror. From our research and testing of the rectifying mirror in Pov-Ray, we identified the rectifying mirror as being optimal for imaging the floor of our test room.  Sorry for the repetition.  JUst making sure everyone is on the same page as us.  Because the rectifying mirror is good at imaging the floor, it will be the foundation for which we add the other mirrors. In Maple we started the numerically solved rectifying mirror on the positive side of the x -axis so that it could be rotated symmerytrically in Pov-Ray. We then gave the rectifying mirror the initial conditions y(0)=1 and set the x- coordinates range to be (x=0.0001..0.70). For the rest of our research the rectifying mirror remained at this fixed point. 

To brigde the gap between the rectifying and forward looking mirror, we added a linear mirror using the equation of a line y=mx+b. The linear mirror was put in between the two mirrors to direct the light rays from the image plane parallel to the forward looking mirror where the light ray will travel a third time back to the image plane. We ensured that the lightrays leaving the image plane after hiting the linear mirror were exactly parallel to the forward looking mirror by giving the line a 45 degree angle slope meaning we used the line equation y=x. To connect all three mirrors, we kept m=1, but ajusted the b from the equation y=mx+b accordingly to test the design. In our final combination we found the equation for our linear mirror to be y=x+0.42155 and the length was 0.15 units given by the range x=0.70..0.85. Through trail and error we realized that the linear mirror worked better at a shorter length, but if it was too short it would not direct enought light rays to the forward looking mirror. 

The third mirror that ended that completed the design of our catadipotric sensor was the forward looking mirror. For our catadoptric sensor we used the forward looking mirror to help with imaging the wall and recieving the light rays directed by the linear mirror. Since the light rays traveling from the linear mirror to the forward looking mirror are parallel to the x-axis, we solved our equation for the function of the forward looking mirror in terms of y. Similar to the computations perviously done for the forward looking mirror, we used the vector method to find F(y). we're not gonna bore you with the excruciating math steps that we have already gone over in previous posts.

So, we went a little bit overboard with our explanation.  But, we can't help it. Once we start talking about math, it's hard to get us to stop.  All we want to say is that we are very excited.  This combination of the rectifying, the 45 degree linear and the forward looking mirror works.  And we found this combination for the first time.  It had never been obtained before.  We capture approximately 160 degree field of view since the remaining of our field of view is blocked by the forward looking mirror.  And here is the doozy, we have minimum distortion in our design.  Just keep in mind that small rings that you see in our Pov-ray picture it has to do with Pov-ray itself and not our design! But, the circle in the floor it's just a gap between our mirrors.

  

Forward Looking Mirror

It's time to learn about another type of mirrors.  we've come a long way from a rectifying mirror to a parabolic, panoramic and now to forward looking mirror. Good job everyone!

As the name forward looking suggests, it reflects a plane directly above its optical axis. Forward looking mirror only works under orthographic projection, where the light rays travel parallel to the mirror's optical axis.  

As explained above, the forward looking mirror reflects a plane above its optical axis such as the plane y=k, where our object is positioned. In other words, the object in on the ceiling. 

Suppose the incoming and outgoing light rays are vectors u and v, respectively. We also know that the normal line n is a vector. Therefore, we use vector method to find an equation for our forward looking mirror.

The first step in solving this problem is to normalize our vectors, by finding a new vector in the same direction as our vectors u and v but in terms of their unit length. In order to do so, we need to find the position vectors of u and v by subtracting their terminal point from their initial point. Then we divide each position vector by its magnitude using the pythagorean theorem.

Using the infinity trick that we used previously with the panoramic mirror helps simplifying the math for normalizing vector v.

 We Add together the two normalized vectors u and v by creating a parallelogram. The sum of these two vectors will be parallel to vector n. We know that parallel vectors have the same slope; thus the slope of the normal vector is the same as the slope of the sum of vectors u and v. Because normal and tangent lines are perpendicular to each other, the slope of n is negative and reciprocal of the slope of the tangent line. Now we know the slope of the tangent line and as a result we can find the equation of the mirror, as simple as that!

Please, don't be intimidated by all these math equations.  We've taught you all the math you need to follow along with us, even if you're not a math person!

We solve the non-linear ordinary differential equation numerically in Maple, which enables us to graph the function of the mirror. To visualize the mirror we create a surface of revolution in Pov-ray by using the points obtained in maple.

Just as a general reminder, the green and black checkers demonstrate the ceiling, while blue and white, red and white, red and black and blue and black are the four surrounding wall in the test room in Pov-ray.