What I would like is to give circles a fill color with a gradient that starts from the middle and then as it moves out to the edges becomes progressively more transparent, giving a blur effect. What is the simplest way of doing this?
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Try setting an appropiately defined java.awt.RadialGradientPaint (using Colors with alpha), then render your circles using that. You may need to translate the graphics coordinate system to get the gradient centered in the circle. (http://docs.oracle.com/javase/7/docs/api/java/awt/RadialGradientPaint.html)
Or just make an image in a graphics program and simply draw the image.

Durandal
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