[tadwseml] King swipe

Given a rectangular grid of points, draw a directed line segment from a point to an orthogonal or diagonal neighbor (one chess king move away).  How many possible line segments are there?

f(x,y)=2*(4*x*y - 3(x+y) + 2)
? f(3,3)
40
? f(3,4)
58
? f(3,5)
76
? f(3,6)
94
? f(4,4)
84
? f(4,5)
110
? f(4,6)
136

This provides a large number of simple input gestures for a touchscreen keyboard.  Every gesture is local, so it is easy to have explanatory annotations.

Other grids possible: consider 12 directions on a hexagonal grid.