| > In my mind abs(x) = x * sign(x) which is why the above formulation seems correct. > This is an interesting idea...how would you define sign(x) in a higher dimension? You could think of the sign as the direction. In the 1-d case, you only have two directions. Positive sign means to the right of the origin, negative sign means to the left of the origin. But in higher dimensional case, you don't get a higher count of directions, instead direction becomes a space. To see this analogy we can rewrite your abs(x) = x * sign(x) as x = abs(x) * sign(x). (Because 1/sign(x) = sign(x) except at 0, where the two equations agree anyway.) Now consider that in higher dimensions, we can write x = ||x||*(x/||x||) for any vector x, where ||x|| denotes the magnitude and the term x/||x|| is the unit vector in direction of x. This term then plays the role of the sign. A simple reflection can then still be done by multiplying this direction term with -1, which in the 2d case reflects at a line through the origin and perpendicular to the vector. I can't comment on the Laplace transform, it's been too long since I used that. |