I have a set of points in the plane which I would like to have "glow". I would like for each point to glow individually and I would also like some increase in the intensity corresponding to an increase in density of the points.

I've come up with a couple ideas for how to do this using DensityPlot but neither are quite what I'm hoping for. I'll describe them below.

I need some points, say

pts = Table[{Re[E^(I t/2 - t/10)], Im[E^(I t/2 - t/10)]}, {t, 1, 50}];

The first idea is to consider an density function like

$$ \frac{1}{\epsilon + \min_{a \in \text{pts}}\operatorname{dist}((x,y),a)}. $$

My code for this is

eps = 1/16; exponent = 1/2;

distfunc1[x_, y_] = 
   1/(eps + Min[
      Table[
         ((x - pts[[k, 1]])^2 + (y - pts[[k, 2]])^2)^(exponent),
         {k, 1, Length[pts]}
      ]
   ]);

Show[
   DensityPlot[distfunc1[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 40],
   Graphics[{PointSize[0.007], Point[pts]}]
]

which produces

The non-differentiability of the density function leads to sharp divisions between the glows. To get around that I considered adding the distances instead of taking the minimum, like

$$ \sum_{a \in \text{pts}} \frac{1}{\epsilon + \operatorname{dist}((x,y),a)}. $$

My definition is

distfunc2[x_, y_] = 
   Sum[
      1/(((x - pts[[k, 1]])^2 + (y - pts[[k, 2]])^2)^(exponent) + eps),
      {k, 1, Length[pts]}
   ];

By varying the parameters eps and exponent I can get parts of what I want. For example with eps = 1/4 and exponent = 1/2 I get nice smooth glows around the outer points but the inner region becomes too "hot":

With eps = 1/2 and exponent = 1/1400 the middle is no longer too hot and has the brightest glow from the density but the outer points no longer have significant idividual glows:

I haven't yet found a way to have a nice strong glow in the center as well as distinct, nontrivial glows for each of the outer points. I appreciate any ideas you may have.

Also, I'm new to Mathematica and I don't really know how ColorFunctions work. Is it easy to increase the range of lights/darks (i.e. increase contrast) in the colorfunction used by DensityPlot to render its pictures? I would like the darkest color to be near-black in the above pictures if possible.

I have a set of points in the plane which I would like to have "glow". I would like for each point to glow individually and I would also like some increase in the intensity corresponding to an increase in density of the points.

I've come up with a couple ideas for how to do this using DensityPlot but neither are quite what I'm hoping for. I'll describe them below.

I need some points, say

pts = Table[{Re[E^(I t/2 - t/10)], Im[E^(I t/2 - t/10)]}, {t, 1, 50}];

The first idea is to consider an density function like

$$ \frac{1}{\epsilon + \min_{a \in \text{pts}}\operatorname{dist}((x,y),a)}. $$

My code for this is

eps = 1/16; exponent = 1/2;

distfunc1[x_, y_] = 
   1/(eps + Min[
      Table[
         ((x - pts[[k, 1]])^2 + (y - pts[[k, 2]])^2)^(exponent),
         {k, 1, Length[pts]}
      ]
   ]);

Show[
   DensityPlot[distfunc1[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 40],
   Graphics[{PointSize[0.007], Point[pts]}]
]

which produces

The non-differentiability of the density function leads to sharp divisions between the glows. To get around that I considered adding the distances instead of taking the minimum, like

$$ \sum_{a \in \text{pts}} \frac{1}{\epsilon + \operatorname{dist}((x,y),a)}. $$

My definition is

distfunc2[x_, y_] = 
   Sum[
      1/(((x - pts[[k, 1]])^2 + (y - pts[[k, 2]])^2)^(exponent) + eps),
      {k, 1, Length[pts]}
   ];

By varying the parameters eps and exponent I can get parts of what I want. For example with eps = 1/4 and exponent = 1/2 I get nice smooth glows around the outer points but the inner region becomes too "hot":

With eps = 1/2 and exponent = 1/1400 the middle is no longer too hot and has the brightest glow from the density but the outer points no longer have significant idividual glows:

I haven't yet found a way to have a nice strong glow in the center as well as distinct, nontrivial glows for each of the outer points. I appreciate any ideas you may have.

Also, I'm new to Mathematica and I don't really know how ColorFunction works. Is it easy to increase the range of lights/darks (i.e. increase contrast) in the color function used by DensityPlot to render its pictures? I would like the darkest color to be near-black in the above pictures if possible.