preRIFS | R Documentation |
preRIFS()
function generates a sample of fractal (prefractal) points in R^n with a random iterated function system (RIFS).
preRIFS(n=10000, Z=R2ngon(), P=rep(1/nrow(Z), times=nrow(Z)), M=rep(1, times=nrow(Z)))
n |
a number of prefractal points. |
Z |
a set of protofractal points. |
P |
a probability distribution of protofractal points. |
M |
a partition coefficients distribution of protofractal points. |
A protofractal set Z
is a discrete or continuous set, which in the iterative process generates a prefractal set X
.
A prefractal set X
is a sample of an attractor (fractal) of a random iterated function system:
X[i,] <- (X[i-1,] + M[z[i]]*Z[z[i],])/(1 + M[z[i]])
,
where the index i in seq(n)
; the index z
corresponds to a random points sample of a protofractal set Z
.
A list with the prefractal ($pre
) and protofractal points ($proto
); distributions of probabilities & coefficients ($distr
); sample of protofractal indexes ($index
).
Pavel V. Moskalev and Alexey G. Bukhovets
Bukhovets A.G. and Bukhovets E.A. (2012) Modeling of fractal data structures. Automation and Remote Control, Vol.73, No.2, pp.381-385, doi:10.1134/S0005117912020154.
Moskalev P.V. and Bukhovets A.G. (2012) The similarity dimension of the random iterated function system. Computer Research and Modeling, Vol.4, No.4, pp.681-691, doi:10.20537/2076-7633-2012-4-4-681-691.
R2ngon, plotR2pre, preRSum0
# Example 1a. Sierpinski triangle, 1st order, p=const, mu=const l <- preRIFS() r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, asp=1, col=r, main="Prefractal points for 3-gon: k=3; p=1/3; mu=1") points(l$pre, pch=46, col=r[l$index]) ## Not run: # Example 1b. Sierpinski triangle, 1st order, p=var, mu=const l <- preRIFS(P=c(2,2,5)/9) r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, asp=1, col=r, main="Prefractal points for 3-gon: k=3; p=(2,2,5)/9; mu=1") points(l$pre, pch=46, col=r[l$index]) # Example 1c. Sierpinski triangle, 1st order, p=const, mu=var l <- preRIFS(M=c(4,4,6)/5) r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, asp=1, col=r, main="Prefractal points for 3-gon: k=3; p=1/3; mu=(4,4,6)/5") points(l$pre, pch=46, col=r[l$index]) # Example 2a. Sierpinski square, 2nd order, p=const, mu=const l <- preRIFS(Z=R2ngon(4,2), M=rep(2,8)) r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, asp=1, col=r, main="Prefractal points for 4-gon: k=8, p=1/8, mu=2") points(l$pre, pch=46, col=r[l$index]) # Example 2b. Sierpinski square, 2nd order, p=var, mu=const l <- preRIFS(Z=R2ngon(4,2), P=2^abs(seq(-3,4))/45, M=rep(2,8)) r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, col=r, asp=1, main="Prefractal points for 4-gon: k=8, p=2^|-3:4|/45, mu=2") points(l$pre, pch=46, col=r[l$index]) # Example 2c. Sierpinski square, 2nd order, p=const, mu=var l <- preRIFS(Z=R2ngon(4,2), M=1.2^abs(seq(-3,4))+0.5) r <- rainbow(n=nrow(l$proto), v=0.9) plot(l$proto, col=r, asp=1, main="Prefractal points for 4-gon: k=8, p=1/8, mu=0.5+1.2^|-3:4|") points(l$pre, pch=46, col=r[l$index]) ## End(Not run)
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