PLOTS/EULER'S EQUATION at master · mlamias/PLOTS · GitHub

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# LINEAR ALGEBRAIC MANIPULATION OF IMAGES IN R:

points = c(1+1i, 1.1+1.1i, 1.2+1.2i, 1.3+1.3i, 1.4+1.4i, 1.5+1.5i,
           1.6+1.6i, 1.7+1.7i, 1.8+1.8i, 1.9+1.9i, 2+2i,
           1.1+1i, 1.1+0.9i, 1.1+0.8i, 1.1+0.7i, 1.1+0.6i, 1.1+0.5i,
           1.1+0.4i, 1.1+0.3i, 1.1+0.2i, 1.1+0.1i, 1.1+0i,
           1.2+0i,1.3+0i,1.3+0i,1.4+0i,1.5+0i, 1.6+0i, 1.7+0i,
           1.8+0i,1.9+0i,2+0i,
           1.7+0.1i, 1.7+0.2i,1.7+0.31i,1.7+0.4i, 1.7+0.5i,
           1.8+0.5i)

points2 = c(2.1+1.9i, 2.2+1.8i, 2.3+1.7i, 2.4+1.6i, 2.5+1.5i,
            2.6+1.4i, 2.7+1.3i, 2.8+1.2i, 2.9+1.1i, 3.0+1.0i,
            2.9+1i,2.9+0.9i,2.9+0.8i,2.9+0.7i,2.9+0.6i,2.9+0.5i,
            2.9+0.4i,2.9+0.3i,2.9+0.2i,2.9+0.1i,2.9+0i,
            2.8+0i,2.7+0i,2.6+0i,2.5+0i,2.4+0i,2.3+0.0i,2.2+0.0i,
            2.1+0i,2.0+0i,1.9+0i,
            2.2+0.1i,2.2+0.2i,2.2+0.3i,2.2+0.4i,2.2+0.5i,
            2.1+0.5i,2.0+0.5i)

# Horizontal and vertical flips:

plot(points, xlim = c(-4.5,4.5), ylim = c(-4.5,4.5), pch = 19,
     col = "tan1",
     ylab="Imaginary axis", xlab="Real axis",
     main="Horizontal and vertical flip")
points(points2, pch=19, col="tan3")
abline(h=0, v=0)

points(- Re(points) + (1i * Im(points)),
       pch=19, col="tan1")
points(- Re(points2) + (1i * Im(points2)),
       pch=19, col="tan3")

points(Re(points) + (-1i * Im(points)),
       pch=19, col="tan1")
points(Re(points2) + (-1i * Im(points2)),
       pch=19, col="tan3")

# Tranlating and scaling:

# Replotting the initial points first...
plot(points, xlim = c(-4.5,4.5), ylim = c(-4.5,4.5), pch = 19,
     col = "tan1",
     ylab="Imaginary axis", xlab="Real axis",
     main="Translating and scaling")
points(points2, pch=19, col="tan3")
abline(h=0, v=0)

# We add -4 + 2.5i to every point, which will move the picture
# to the left 4 points and up 2.5:
points((-4 + 2.5i) + c(points, points2), pch=19, col=5)

# Now we enlarge the image 2x at the same time that we move it down
# and to the left:

points(2 * ((-2.8 + -1i) + c(points, points2)), pch=19, col=4)

# Inverting:
# We start off replotting:

plot(points, xlim = c(-4.5,4.5), ylim = c(-4.5,4.5), pch = 19, col = 10,
     ylab="Imaginary axis", xlab="Real axis",
     main="90 degree rotation and inverting")

points(points2, pch=19, col=4)
abline(h=0, v=0)

# We multiply all points by -1, meaning Re. and Im. parts alike:

points(-1 * points, pch = 19, col = 10)
points(-1 * points2, pch = 19, col = 4)

# 90 degree rotation:
# Multiplying by i will turn the Re. part into the Im., and the
# Im. part into Re.:

points(1i * points, pch = 19, col = 10)
points(1i * points2, pch = 19, col = 4)


# We can now rotate at smaller increments:
# Plotting first the initial object:

combo <- c(points,points2)
plot(combo, xlim = c(-5,5), ylim = c(-5,5), pch = 19, col = 2,
     main="Rotations", xlab= "Real Ax", ylab="Im. Ax.")

abline(h=0, v=0)

# The rotation formula that follows partitions the 2pi circumference
# in 200 parts, and then rotates the image at different increments
# in these partitions. Also, it enlarges the image 1.5x:

for (n in seq(20, 180, 40)){
    rot_points <- 1.5 * exp(n * 1i * (2*pi / 200)) * combo
    points(rot_points, pch = 19, col="steelblue4")
}


# Stops: Fractions of the circle: 2pi / n

# Euler's equation: e^i*x = cos (x) + i sin (x)

f <- function(stops){counts <- 1:stops; exp(counts * 1i * (2*pi / stops))}

stops = 20
plot(f(stops), pch = 20, col='tan4', main = "Euler's Identity",
     ylab = "Imaginary axis", xlab = "Real axis")

stops =  1000
plot(f(stops), pch = 19, col = 'tan4', cex = 0.3,
     main = "Euler's Identity",
     ylab = "Imaginary axis", xlab = "Real axis")


fun <- function(stops){
    counts <- 1:stops
    (0.1 * counts) * exp(counts * 1i * (2*pi / stops))
}

plot(f(stops), pch = 19, col = 'tan4', cex = 0.1,
     xlim=c(-10,10), ylim=c(-10,10),
     main = "Euler's Identity",
     ylab = "Imaginary axis", xlab = "Real axis", type = 'n')

for(i in seq(50, 100, 5)){
    points(fun(i), pch = 20, col = i)
    }

fun <- function(stops){
    counts <- 1:stops
    (1 - (0.05 * stops)) * exp(counts * 1i * (2*pi / stops))
}

plot(f(stops), pch = 19, col = 'tan4', cex = 0.1,
     main = "Euler's Identity",
     ylab = "Imaginary axis", xlab = "Real axis", type = 'n')

for(i in 20:40){
    points(fun(i), pch = 19, cex = 0.5, col = i)
}