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.

Panoramic Mirror with Infinity Trick

Not to sound lazy or anything but the numeric solution for the panoramic mirror in the last post was just too much work!  That is why we tried to solve it analytically, this time.
So, we still have the same panoramic mirror as before, where the object is positioned on a surrounding wall with the coordinates (k, ax+b). The image plane is below the x-axis and is indicated by y=-k.

But, here is the time to use a trick.  We know that "a" is an arbitrary scaling factor and the bigger the "a", the bigger the field of view.  Therefore, it shouldn't matter whether it is ax+b or kax+b or gax+b.  "a", "ka" and/ or "ga" are all scaling factors.  So, for the sake of our trick, let's say it is kax+b. 

Now, once again, we draw the tangent line to the mirror at point (x, F(x)) and we have:

(1)

 (2)

Using the tangent double-angle formula we have:  

 (3)

Here is where the trick comes into the picture.  Suppose k is going to infinity and find the limit of tangent of 2theta:

 (4)

It looks so cool, doesn't it?

By equating the (2) and the (4) equations, we get:

YAY! We got our differential equation.  Now, by solving this differential equation in maple and finding the surface of revolution in Pov-ray we obtain the same results as we did previously :)