Some useful functions for art (2024)

This post is to keep track of some functions of mine that I find useful in making mathematical art.

in python

import numpy as npimport matplotlib.pyplot as pltdef harmonica(NNN=1000, waves=23, ratio=24/23): thetas = np.linspace(0, 1, num=NNN) * 2 * np.pi * waves return np.column_stack((np.cos(thetas), np.sin(ratio * thetas)))fig, axs = plt.subplots(2, 2)for ax, ratio, waves in zip(axs.flatten(), [1, 4/3, 8/7, 24/23], [0.95, 2.95, 6.95, 22.95]): harmonics = harmonica(ratio=ratio, waves=waves) ax.plot(harmonics[:, 0], harmonics[:, 1], color='black', linewidth=0.5) ax.set_xticks([]) ax.set_yticks([]) ax.set_aspect('equal') ax.axis('off') # Turn off the axes and bounding box

## [<matplotlib.lines.Line2D object at 0x126bf5350>]## []## []## (-1.0999991692927558, 1.0999999604425121, -1.0999981210196357, 1.0999997033188356)## [<matplotlib.lines.Line2D object at 0x126c27390>]## []## []## (-1.0999997923231684, 1.099999990110627, -1.0999357707940272, 1.0999334852952614)## [<matplotlib.lines.Line2D object at 0x126c80950>]## []## []## (-1.099991225665797, 1.0999995821745618, -1.099998840017093, 1.0999998168447835)## [<matplotlib.lines.Line2D object at 0x126cb2750>]## []## []## (-1.1, 1.1, -1.0999998977880034, 1.0999993526573957)

plt.tight_layout()plt.savefig("harmonica_plots.png", dpi=300)plt.show()

This function computes the distance traveled along a curve as a percentage of total distance traveled, which can be used to create a new curve where the points are evenly spaced. Works for a curve that is either in 2D or 3D space. One can make an evenly spaced vector by using linear interpolation for a lookup table is combination of distance traveled and the x (or y vector) and one “looks up” a set of numbers that are evenly spaced on the unit interval. In the pictures below the “dots” cluster in the corner in the graph on the right as they are not evenly spaced.

traveled <- function(xy) { NN <- nrow(xy) if (ncol(xy) == 2) { distanceBetween <- c(0, sqrt(rowSums((xy[1:(NN - 1), 1:2] - xy[2:(NN), 1:2])^2))) } else { distanceBetween <- c(0, sqrt(rowSums((xy[1:(NN - 1), 1:3] - xy[2:(NN), 1:3])^2))) } traveled <- c*msum(distanceBetween) / sum(distanceBetween) return(traveled)}# Example:NNN0 <- 500xy0 <- harmonica(NNN = NNN0, ratio = 4 / 3, waves = 2.95)travel <- traveled(xy0)xy <- cbind( approx(travel, xy0[, 1], seq(0, 1, length = NNN0), rule = 2)[[2]], approx(travel, xy0[, 2], seq(0, 1, length = NNN0), rule = 2)[[2]])par(pty = "s", ann = F, mai = 0 * c(1, 1, 1, 1), mfrow = c(1, 2))plot(xy, typ = "l", axes = F)points(xy, pch = 20, col = rgb(1, 0, 0, .5))plot(xy0, typ = "l", axes = F)points(xy0, pch = 20, col = rgb(1, 0, 0, .5))

This function conveniently transforms a sin curve to a given top and bottom and controls where the sin curve starts zero and the number of waves.

dougsSign <- function(topBottom, NNN, waves = 2 * pi, start0 = 0) { thetas <- seq(0, 1, length = NNN) * waves wave <- (topBottom[1] - topBottom[2]) * ((sin(thetas + start0) + 1) / 2) + topBottom[2] return(cbind(thetas, wave))}dY <- 1 / 4par(pty = "s", ann = F)plot(NULL, xlim = c(0, 2 * pi), ylim = c(-1, 1), axes = F)start <- 32stop <- 18for (i in seq(start, stop, -2)) { xy <- dougsSign((start - i) * dY / 2 + c(-1, -1 + dY), 1000, waves = i * pi) lines(2 * xy[, 1] / i, xy[, 2])}

This function I call polly as I have cousin with that name. It creates a regular polygon using polar coordinates with a default radius of 1. If starts is set to 0 (or pi/2) is starts at c(0,1) (or c(1,0)), and it moves clockwise.

polly = function(sides=4,start=0,r=1){ thetas=seq(0,1,length=sides+1)*2*pi res=r*cbind(sin(start+thetas),cos(start+thetas)) return(res)}

par(pty='s',ann=F,mai=0*c(1,1,1,1),mfrow=c(1,2))p1=polly()p2=polly(start=pi/6)p3=polly(start=pi/3) plot(p1,t="l",axes=F)lines(p2)lines(p3)p1 = polly(12)plot(NULL, xlim = c(-1,1),ylim = c(-1,1), axes=F)text(p1[,1],p1[,2],1:12)for (i in 1:35) { lines(rbind(p1[(i %% 12) + 1,],p1[(i+5) %% 12 + 1,])) }

#bigger stars

par(pty='s',ann=F,mai=0*c(1,1,1,1),mfrow=c(1,2))p1 = polly(11)plot(NULL, xlim = c(-1,1),ylim = c(-1,1), axes=F)text(p1[,1],p1[,2],1:11)for (i in 1:35) { lines(rbind(p1[(i %% 11) + 1,],p1[(i+4) %% 11 + 1,])) }p1 = polly(7)plot(NULL, xlim = c(-1,1),ylim = c(-1,1), axes=F)text(p1[,1],p1[,2],1:7)for (i in 1:14) { lines(rbind(p1[(i %% 7) + 1,],p1[(i+2) %% 7 + 1,])) }

The function is named in honor of Forbes Whiteside who taught art at Oberlin College. He used to say tha the way an artist drew a perfect circle was to lightly draw an imperfect circle and then draw another correcting the mistakes of the first one and repeat and the collection of light circles would become a dark perfect circle reinforced by the mistakes or something like that.

whiteside = function(rho = .9, sigmae = .02, theEnd = 100, width = 1, white = rgb(1, 1, 1, .15), white2 = rgb(1, 1, 1, 1), scale = 1, scaleY = 1, rTheta = 0, shiftV = c(0, 0), fill = F) { start = 0 end = theEnd * 64.1 e = rnorm(end) mean = 1 a = mean + e * sqrt(sigmae ^ 2 / (1 - rho ^ 2)) for (i in 2:length(e)) { a[i] = mean * (1 - rho) + rho * a[i - 1] + sigmae * e[i] } sqrt(sigmae ^ 2 / (1 - rho ^ 2)) if (fill == T) { thetas = 2 * pi * seq(0, 1, .01) circle = cbind(scale * sin(thetas), scale * scaleY * cos(thetas)) circle = rotate(circle, rTheta) circle = shift(circle, shiftV) polygon(circle, col = white2, border = NA) } thetas = (1 / 64.1) * seq(1:end) * 2 * pi for (j in 1:theEnd) { start = (j - 1) * 65 + 1 end = start + 65 circle = cbind(a[start:end] * scale * cos(thetas[start:end]), a[start:end] * scale * scaleY * sin(thetas[start:end])) circle = rotate(circle, rTheta) circle = shift(circle, shiftV) lines(circle, typ = "l", col = white, lwd = width) }}rotate=function (xy,theta,about=c(0,0)) { rows=nrow(xy) center=t(matrix(rep(about,rows),2,rows)) rotate=rbind(c(cos(theta),sin(theta)),c(-sin(theta),cos(theta))) rxy =(xy-center) %*% rotate +center return(rxy)}shift=function(xy,shifted){ rows=nrow(xy) xynew=xy+t(matrix(rep(shifted,rows),2,rows)) return(xynew)}par(pty="s",ann=FALSE,mfrow=c(1,1),bg="grey70",mai=.1*c(1,1,1,1)) plot(NULL, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),axes=FALSE) whiteside( rho = .98, sigmae = .015, theEnd = 500, white = rgb(1, 1, 1, .2), white2 = rgb(1, 1, 1, 1), width = 1, fill = F )

Some useful functions for art (2024)

in python

References