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Recusively calculating curve length question
Posted:
Nov 23, 2012 5:16 PM
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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
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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.
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