Simulating a spider web

 

A friend shared a video of a spider making web.

Found the equations for drawing different components of the web.

The figure above was drawn by the following code in R.

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# Code based on material provided by Andrew Chambers

# https://ibmathsresources.com/2021/09/10/weaving-a-spider-web/

# 30 Sep 2022

graphics.off()

# First make a graph paper where the web shall be drawn

plot(1,1, type="n", pch=".", xlim=c(-15,15), ylim=c(-15, 15), 

     xlab="", ylab="", axes=F)

for(i in (-15):15) {

  if(i%%5==0) {

    mycol = "gray80"

  } else {

    mycol="gray90"

  }

  abline(h=i, v=i, col=mycol)

}

# Straight lines passing through sharp points

# formula for getting the lines

# Note: In the tutorial, this was drawn at the end

# However it is drawn first here to make it consistent

# with the video of actual spider's work today

get_slope = function(p) {

  numr = sin(9*pi*p/5)-9*sin(pi*p/5)

  denr = cos(9*pi*p/5)+9*cos(pi*p/5)

  return(numr/denr)

}

x = seq(-15,15,by=0.001)

for(p in 1:5) {

  y = get_slope(p)*x

  points(x,y, type="p", pch=".", col="black", cex=2)

}

mtext(text="Step 1: Straight lines", col="black", side=1, cex=2)

# Archimedean spiral, modified from the tutorial

# In the tutorial it was drawn after the hypocycloid

t <- seq(0,7, len=1000)  # the parametric index

# Then convert ( t, 2*pi*t ) to rectilinear coordinates

x = (t)* cos(2*pi*t) 

y = (t)* sin(2*pi*t)

points(x,y, type="l", col="blue", lwd=3)

mtext(text="Step 2: Spiral", col="blue", side=2, cex=2)

# hypocycloid

get_x = function(tx, n=9) {

  cos(tx) + n*cos(tx/9)

}

get_y = function(tx, n=9) {

  sin(tx) - n*sin(tx/9)

}

tx = seq(from=1, to=100, by=0.01)

n = 9

x = get_x(tx, n=n)

y = get_y(tx, n=n)

for(n in 9:13) {

  x = get_x(tx, n=n)

  y = get_y(tx, n=n)

  points(x,y, type="p", pch=".", col="red", cex=2)

}

mtext(text="Step 3: Outer lines", col="red", side=3, cex=2)

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