Spread, wedge, dot, part 1

Let’s read “Spread polynomials, rotations and the butterfly effect”  by Norman Wildberger and Shuxiang Goh. The link is on Arxiv.org, here: http://arxiv.org/pdf/0911.1025.pdf

If you have two lines with equations like so:

l_1 \equiv a_1x + b_1y + c = 0

l_2 \equiv a_2x + b_2y + c = 0

then you can describe the ‘separation between the lines’ as Spread, s,

s \equiv \dfrac{(a_1b_2-b_1a_2)^2}{(a_1^2+b_1^2)(a_2^2+b_2^2)}

But what if we imagine that a_1 and b_1 are the coordinates of a vector, v_1, and the same for a_2, b_2 and a vector v_2?

v_1 = [a_1,b_1] , v_2= [a_2,b_2]

Then we could describe spread like so:

s \equiv \dfrac{(v_1 \wedge v_2)^2}{(v_1 \cdot v_1)(v_2 \cdot v_2)}

In other words, spread is a ratio between wedges and dot products.

But the funny part is that our wedges and dot products here are not in the space that is directly related to our original lines l_1 and l_2. It is as though we have two different worlds we speak of. What is in one world is a reflection of the conditions in the other, and vice versa.



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