plotting - How to create a Disk with smoothly changing coloring?

I want to have a disk. But this disk shouldn't have a fixed color; instead, I'd like it's color to fade smoothly.

Until now, I came up with two ideas how to achieve this.

Idea #1:

smoothlyFadingColorDisk1[r_] := 
 DensityPlot[Max[0, r - Sqrt[x^2 + y^2]], {x, -r, r}, {y, -r, r}, 
  ColorFunction -> (Opacity[#, Purple] &)]

Problems:

It displays axes, and Axes->False won't remove them. This is meant to be just a part of a more complex graphics, so the axes are here completely inaproppiate. (blocking problem)

This makes no sense semantically. I want a Disk (a simple image), not a plot.

Idea #2:

smoothlyShiftedColorDisk2[r_, n_] := 
 Graphics[Flatten@
   Prepend[{Thickness[1/(2*n - 1)]}]@
    Table[{Opacity[(n - i + 1)/n, Purple], 
      Circle[{0, 0}, (i - 1/2)*r/(n - 1/2)]}, {i, 1, n}]]

Problems:

It looks bad at some values of n and horrible at others. This seems to be because for some reason (inaccurate floating point calculations? my mistake in the maths?) the concentric circles are overlapping, and they shouldn't. They should be ideally next to each other. (blocking problem)

I have to manually set the number of colors used (the n variable, number of concentric cirlces). And I'd like Mathematica to take care of this for me, as happens in the first example.

So - neither of these two ideas acheves what I am trying to achieve. Er... How to do this properly?

I have a feeling that this should be possible to achieve in a much simpler way than I was trying to do this...

Comments

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I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

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equation solving - Invert and fit implicitly defined curve

I need to fit an implicitly defined curve. I thought I could get some data out of Solve , and then using FindFit . Therefore, I would like to find the relation the parametric curve defined by $F(x,y)=0$: Solve[-(1/2) + 1/2 (0.41202 BesselK[0, 0.1 Sqrt[x^2 + y^2]] + (0.101483 x BesselK[1, 0.1 Sqrt[x^2 + y^2]])/Sqrt[x^2 + y^2]) == 0, y] But I can't get an output: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. >> Edit: In particular, I would like to fit the data coming from the curve with the expression of another curve, and not with a function $f(x)$. In particular, since this clearly looks like a cardioid , I would like it to fit to something like it. What other strategies could I try?

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