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Length of a spiral (parametric)

by Mounty
Tags: length, parametric, spiral
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Mounty
#1
Feb2-05, 07:48 AM
P: 5
given the parametric eqns for a spiral

x=kt cos t
y=kt sin t

where k is a constant

give a function of 't' that calculates the length of the spiral.
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dextercioby
#2
Feb2-05, 07:50 AM
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Do you know how to calculate the length of curves?In general,what formula need you apply...?

Daniel.
Mounty
#3
Feb2-05, 07:54 AM
P: 5
Nope don't know them....

I'm plotting a spiral by increasing t+=0.1

I need to know the length of the spiral plotted as a function of t,x and y

basically I want to distributes points along the circle spaced evenly by distance....

this I can only do if I know how far I've currently plotted.....

dextercioby
#4
Feb2-05, 07:59 AM
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Length of a spiral (parametric)

Should i understand that u're not looking for an anlytical solution and u may never heard of first kind curvilinear integrals...?

You should have said from the beginning what kind of sollution u were looking for...

Daniel.
Mounty
#5
Feb2-05, 08:02 AM
P: 5
Dunno.....I thought I'd made it fairly clear, my apologies if not. Never heard of curvilinear integrals....sorry. Done loads of googling on this subject but didn't find anything that gave a solution to my particular problem.

I just need a formula along the lines of

lenght of spiral = some function of t

An explanation of how it was derived would be great....but not vital....
dextercioby
#6
Feb2-05, 08:12 AM
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In parametric coordinates,it's this formula that gives the length of a curve:
[tex] L_{C}=\int_{t_{1}}^{t_{2}} \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}} \ dt [/tex](1)

So you see,it's a Riemann definite integral (it's not really a function of "t",as the "t" is getting integrated along the curve)...
The general formula from which one finds (1) is
[tex] L_{C}=\int_{C} dl [/tex](2)
,where "C" is the curve whose length u wanna find & "dl" is the line element along the curve...

So using (1) and 2 points along the curve (which means chosing 2 distinct values of the parameter 't'),u can find the length,expressed as a real number.

Daniel.
Mounty
#7
Feb2-05, 08:21 AM
P: 5
Thanks for the info, but I don't think it really solves my problem -

The curve is being plotted realtime in cartesian spac, so I need to know when I've plotted a distance of N units along the curve as the curve is generated.

If I always used t=0 as the first distinct parameter then could this be done?Sorry for my complete lack of understanding here but it's bloody ages since I've done any calculus....all I really want is an equation I can chuck some numbers in and get an answer from....

:(
dextercioby
#8
Feb2-05, 08:32 AM
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To properly use formula #1,u need to supply the input parameter 't' with 2 values corresponding to the 2 ends of the curve...If u've chosen the first to be "0",it's okay.You still need another value,however...

Did u do the differentiations & squarings correctly...?

Daniel.
Mounty
#9
Feb2-05, 08:58 AM
P: 5
man this is embaressing but I remember so little of this stuff

does 't cos t' differentiate to -t sin t?

and

t sin t differentiate to t cos t?

If you could post the final eqn it'd be really useful....I've got a client deadline to meet....
dextercioby
#10
Feb2-05, 09:17 AM
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Okay.Your differential calculus is a bit rusty.So let's use the PRODUCT RULE:

[tex] \frac{dx}{dt}=x'=k(t\cos t)'=k\cos t-kt\sin t [/tex] (1)

[tex] \frac{dy}{dt}=y'=k(t\sin t)'=k\sin t+kt\cos t [/tex] (2)

Now square (1) & square (2),add the results,use the fundamental identity of circular trigonometry:
[tex] \sin^{2}t+\cos^{2}t =1 [/tex] (3)

and finally take square root of the everything u've obtained so far.

Plug everything in the integral.

Daniel.
E=MC**2
#11
Jan15-07, 05:13 AM
P: 1
I would like to determine X and Y coordinates along a spiral as posed below:

Given: Internal diameter of spiral = 500 meters
External diameter of spiral = 7000 meters
Pitch of spiral = 30 meters

Assuming the center of the spiral is 0,0 and I start at 0,250, how do I calculate the X and Y coordinates for points every 30 meters along the spiral?


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