Brief analysis & simplification of terms

Now that we've seen the different types of matrices we are in a position where we can combine this knowledge with what we know to be the properties of a zoom lens.  This will give us an idea as to what needs to happen in order for us to develop an effective theoretical model.

In Focus?
The factor determinig whether or not we are in focus is the bottom left entry of the system matrix.  Though this is only part of the system matrix, it might as well be the whole thing because it still looks pretty intimidating at first glance.
However, this monstrosity can be simplified.  Our first step to understanding this is recognizing what is and what isn't a constant.  It turns out that only x and y are variables and everything else is constant... sort of.  Once we've chosen our lenses, the focal lengths are fixed.  No matter how near or far the object is, the focal lengths will not change because they are characteristics of the lenses only.  Also, if we were to physically construct the lens and put it up to our eye (or attach it to a camera), the distance from the rear lens to our eye (or the film) would not change.  Finally, once we focus on a distant object, the distance from the front of the lens to the object remains almost constant.  It is not really constant because the length of our zoom lens will vary a little.  What is important to recognize here is that we choose to limit our lens to a maximum length of one meter.  This way we not only make sure that it would not be too large to carry, but also that the length of the zoom lens will be small relative to the distance from the object.  After all this simplification, we have a function in terms of two variables only:  x and y.  Setting this equal to zero, we can rearrange this to find x in terms of y.

Magnified?
Having answered the focus question, we are left with the question of magnification.  Just as the bottom left entry told us whether or not the object was in focus, the top right term tells us the power of the system.
You should be able to see that we haven't introduced anything new to this term; thus, it is no different from the bottom left term in that only x and y are variables.  If we find solutions for the bottom left term and then plug them in to the top right term, we will end up with a number representing the power (magnification) of our lens.  We hope to find that this number is positive - indicating an upright image - and that we will have a good range of values depending on x and y.


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