Consider two point masses P₁ and P₂ in the plane moving with instantaneous velocities:

Consider the Euclidian distance:

These points will be approaching (resp. moving away from) each other if the time derivative of the above function is negative (resp. positive), i.e.: 

Given a point C in the plane, specify n₁, n₂ as infinitesimal flows over a non-linear rotational field:

The question we have ask is, given fixed mass locations P₁ and P₂, what is the locus of C points which, under the above flow, causes P₁ and P₂ to approach each other?

The answer, which can be derived analytically, yields the following interesting result. Let L₁ be a straight line passing thru P₁ and P₂. Let L₂ be a straight line perpendicular to L₁, passing thru the mid point of segment P₁P₂. These two lines will divide the plane into four quadrants Q1 thru Q4. P₁ will approach (resp. move away from) P₂ after a counterclockwise flow about C if C lies in Q2 or Q4 (resp. Q1 or Q3).

This result is illustrated below for P₁=(-1,-1) and P₂=(1,1). The coloring corresponds to the dot product (P₁-P₂).(v₁-v₂), mapped thru the colormap shown below the picture. Black corresponds to zero change in distance; bright red (resp. blue) corresponds to large decreases (resp. increases) in relative distance. The green lines mark the no-change-in-distance regions. 

Another interesting test is that of "area shrinkage" -- given the point masses and the associated triangle connecting them, how does the position of C affect the area of the new triangle after an infinitesimal flow about C? Since the area test is a simple (and linear) cross-product, the analysis is simple. A good visualization of this phenomenon is shown below (here the blue hue has been replaced by green). Also shown are level curves for C positions which result in the same shrinkage/expansion of the triangle area.