## Tuesday, January 1, 2013

### Crossing point theorem

I found this:

If you know the endpoints of the line, then A for example is the small lines furthest right x coordinate minus it's left x coordinate. The arrow points to what is to subtract what as for vectors.
E for another example is the y coordinate of the bottom right vertex minus the y coordinate of the small lines left vertex.
Proof:
Above is a parameterization of two lines A and B with respect to U and T.
Here is the general solution as Maple solves it. But I'd like to find a better way to write it.
The denominator of t above can be rewritten as:, immediately below it is the the expansion of it.
The numerator of t as:

Now for a second consider this big fraction to be called W. I'll be plugging this back in for t in the X(T) equation but also rewrite the result of that using this identity:
Which together gives:

Now the nice thing about writing it this way is every term involves only an x or a y and there is one minus sign inside every parenthesis so these can be thought of as distances. Zy works almost exactly the same but with a couple differences.

These together with variable renaming give the result.

Where as I explained the end of the arrow points to the term that is subtracting the other term as it would be for vectors. For example the A term in the final equation came from x(2,t) - x(1,t) which is the beginning and end x coordinate of the first line.