Date: Nov 23, 2012 5:16 PM
Author: Brad Cooper
Subject: Recusively calculating curve length question

I have spent quite a few hours trying to write a recursive routine
to calculate the length of a curve by adding up small secants.

I use a straight forward summation to calculate a correct approx. length

===============================================

Set up the curve using X(t) = E^t and Y(t) = sin(t)

X := t -> E^t:
Y := t -> sin(t):

Use the interval [a, b] where a = -1.0 and b = 1.0


Firstly, use a straight forward summation to get the correct length:

a := -1.0:
b := 1.0:

n := 50:            //use 50 secants
h := (b-a)/n:     //width of each secant


Now calculate the approx. length of the curve

sum(sqrt( (X(a+h*(i))-X(a+h*(i-1)))^2 +
          (Y(a+h*(i))-Y(a+h*(i-1)))^2 ), i=1..n);

Answer for length is 2.99758079

======================================================

Now set up a recursive routine to calculate the curve length:
(One of many variations)

secantLen := proc(a, b, tolerance)
begin
 L := sqrt( (X(b)-X(a))^2 + (Y(b)-Y(a))^2 ):
 c := (a+b)/2:

 if L > tolerance then
    L  := secantLen(a, c, tolerance) + secantLen(c, b, tolerance):
 end_if:

return(L):
end_proc:


Call the recursive routine:

a := -1.0:
b := 1.0:
tolerance := 0.05:

secantLen(a, b, tolerance):
float(%);

This gives 5.984159884 (incorrect)

It is difficult to debug. Any help appreciated.

Cheers.