# Length of a spiral (parametric)

by Mounty
Tags: length, parametric, spiral
 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.
 Sci Advisor HW Helper P: 11,927 Do you know how to calculate the length of curves?In general,what formula need you apply...? Daniel.
 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.....
 Sci Advisor HW Helper P: 11,927 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.
 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....
 Sci Advisor HW Helper P: 11,927 In parametric coordinates,it's this formula that gives the length of a curve: $$L_{C}=\int_{t_{1}}^{t_{2}} \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}} \ dt$$(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 $$L_{C}=\int_{C} dl$$(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.
 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.... :(
 Sci Advisor HW Helper P: 11,927 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.
 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....
 Sci Advisor HW Helper P: 11,927 Okay.Your differential calculus is a bit rusty.So let's use the PRODUCT RULE: $$\frac{dx}{dt}=x'=k(t\cos t)'=k\cos t-kt\sin t$$ (1) $$\frac{dy}{dt}=y'=k(t\sin t)'=k\sin t+kt\cos t$$ (2) Now square (1) & square (2),add the results,use the fundamental identity of circular trigonometry: $$\sin^{2}t+\cos^{2}t =1$$ (3) and finally take square root of the everything u've obtained so far. Plug everything in the integral. Daniel.
 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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