Return arc length of every rotation of an Archimedean Spiral given arm spacing and total length

Question

I want to calculate the length of every full rotation of an Archimedean Spiral given the spacing between each arm and the total length are known. The closest to a solution I've been able to find is here, but this is for finding an unknown length.

I can't interpret math notation so am unable to extrapolate from the info in the link above. The closest I've been able to achieve is:

Distance between each spiral arm:

ArmSpace <- 7

Total length of spiral:

TotalLength <- 399.5238

Create empty df to accommodate TotalLength (note that sum(df[,2]) can be > TotalLength):

df <- data.frame(matrix(NA, nrow=0, ncol=2))
colnames(df) <- c("turn_num", "turn_len_m")
df[1,1] <- 0 # Start location of spiral
df[1,2] <- pi*1/1000

Return length of every turn:

i <- 0
while(i < TotalLength) {
  df[nrow(df)+1,1] <- nrow(df) # Add turn number
  df[nrow(df),2] <- pi*(df[nrow(df)-1,2] +
                          (2*df[nrow(df),1])*ArmSpace)/1000
  i <- sum(df[,2])
}

An annotated example explaining the steps would be most appreciated.

Answer 1

I used approximation Clackson formula

t = 2 * Pi * Sqrt(2 * s / a)

to get theta angle corresponding to arc length s.

Example in Delphi, I hope idea is clear enough to implement in R

var
  i, cx, cy, x, y: Integer;
  s, t, a, r : Double;
begin
  cx := 0;
  cy := 0;
  a := 10;           //spiral size parameter
  Canvas.MoveTo(cx, cy);
  for i := 1 to 1000 do begin
    s := 0.07 * i;   //arc length
    t :=  2 * Pi * Sqrt(2 * s / a);   //theta
    r := a * t;                       //radius
    x := Round(cx + r * cos(t));      //rounded coordinates 
    y := Round(cy + r * sin(t));
    Memo1.Lines.Add(Format('len %5.3f theta %5.3f r %5.3f x %d y %d', [s, t, r, x, y]));
    Canvas.LineTo(x, y);
    if i mod 10 = 1 then   //draw some points as small circles
       Canvas.Ellipse(x-2, y-2, x+3, y+3);
  end;

Some generated points

len 0.070 theta 0.743 r 7.434 x 5 y 5
len 0.140 theta 1.051 r 10.514 x 5 y 9
len 0.210 theta 1.288 r 12.877 x 4 y 12
len 0.280 theta 1.487 r 14.869 x 1 y 15
len 0.350 theta 1.662 r 16.624 x -2 y 17
len 0.420 theta 1.821 r 18.210 x -5 y 18

Link gives exact formula for ac length,

s(t) = 1/(2*a) * (t * Sqrt(1 + t*t) + ln(t + Sqrt(1+t*t)))

but we cannot calculate inverse (t for given s) using simple formula, so one need to apply numerical methods to find theta for arc length value.

Addition: length of k-th turn. Here we can use exact formula. Python code:

import math
def arch_sp_len(a, t):
    return a/2 * (t * math.sqrt(1 + t*t) + math.log(t + math.sqrt(1+t*t)))

def arch_sp_turnlen(a, k):
    return arch_sp_len(a, k*2*math.pi) - arch_sp_len(a, (k-1)*2*math.pi)

print(arch_sp_turnlen(1, 1))
print(arch_sp_turnlen(1, 2))
print(arch_sp_turnlen(10, 3))