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

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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.