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Length of a spiral (parametric) 
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#1
Feb205, 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. 


#2
Feb205, 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. 


#3
Feb205, 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..... 


#4
Feb205, 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. 


#5
Feb205, 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.... 


#6
Feb205, 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. 


#7
Feb205, 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.... :( 


#8
Feb205, 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. 


#9
Feb205, 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.... 


#10
Feb205, 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 tkt\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. 


#11
Jan1507, 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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