## 4.5-3
##
save_plots = FALSE

if( save_plots ){
    fn = '../../WriteUp/Graphics/Chapter4/chap_4_sec_5_ex_3_regions.eps'
    postscript(fn, onefile=FALSE, horizontal=FALSE)
}

a1 = seq(-1, +1, length.out=200)

## Part (a):
##
a2_top = 1-abs(a1)
a2_bottom = -(1-abs(a1))

a2_min = min(c(a2_bottom, a2_top))
a2_max = max(c(a2_bottom, a2_top))

plot(a1, a2_top, 'l', col='blue', ylim=c(a2_min, a2_max), xlab='a1', ylab='a2')
lines(a1, a2_bottom, 'l', col='blue')

## Part (b):
##
a2_top = (1-abs(a1)^(3/2))^(2/3)
a2_bottom = -(1-abs(a1)^(3/2))^(2/3)
lines(a1, a2_top, 'l', col='red')
lines(a1, a2_bottom, 'l', col='red')

## Part (c):
##
a2_top = (1-abs(a1)^2)^(1/2)
a2_bottom = -(1-abs(a1)^2)^(1/2)
lines(a1, a2_top, 'l', col='green')
lines(a1, a2_bottom, 'l', col='green')

## Part (d):
##
a2_top = (1-abs(a1)^4)^(1/4)
a2_bottom = -(1-abs(a1)^4)^(1/4)
lines(a1, a2_top, 'l', col='purple')
lines(a1, a2_bottom, 'l', col='purple')

## Part (e):
##
p = 40
a2_top = (1-abs(a1)^p)^(1/p)
a2_bottom = -(1-abs(a1)^p)^(1/p)
lines(a1, a2_top, 'l', col='black')
lines(a1, a2_bottom, 'l', col='black')
grid()

legend('center', legend=c('1-norm', '3/2-norm', '2-norm', '4-norm', 'Infinity-norm'), col=c('blue', 'red', 'green', 'purple', 'black'), lty=1)

if( save_plots ){
    dev.off()
}