[R] How to get transparent colors to sum to complete opacity?

[Prof Brian Ripley]

Look up 'alpha-blending' to see how this works.  And remember that the 
sRGB colorspace used has non-linear transformations too.

Superposing a finite set of semi-transparent dots will never give an 
opaque one (and that is true of physical dots too).

On 17/12/2012 08:55, Andrew Crane-Droesch wrote:
> Dear List,
>
> I want to use transparency in R to represent downweighting of
> observations based on clusters (repeated observations in a dataset).
> Some clusters will have identical covariate values in a parameter space
> -- in the 2D x,y case, these represent a bunch of semi-tranparent dots
> in the same place.  I'd like these overlapping dots to be completely
> opaque.  In other cases, the clusters don't have overlapping covariates,
> so when these dots are scattered all around, I want them to be somewhat
> transparent.
>
> But it seems clear that transparency isn't additive.  For example, four
> dots with transparency set to .25 don't add to complete opacity:
>
> x = c(1,1,1,1)
> y = c(1,1,1,1)
> w = .25
> plot(x,y,pch=16,col=rgb(0,0,1,.25,maxColorValue=1),cex=3,xlim=c(.8,2))
>
> My question is the following: what function would I transform "w" by to
> make it so that 4*f(w) = fully opaque?
>
> The following would suggest f(w) = w^.5, but I'd appreciate if someone
> could confirm for applications outside this little example, and give me
> a sense of how this all works, and is intended to work.
>
> x = c(1,1,1,1)
> y = c(1,1,1,1)
> f = 0
> plot(x,y,pch=16,col=rgb(0,0,1,(.25),maxColorValue=1),cex=3,xlim=c(.8,2))
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
> x=x+.1; f=f-.1
> points(x,y,pch=16,col=rgb(0,0,1,(.25)^(1+f),maxColorValue=1),cex=3)
>
> Thanks,
> Andrew
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595