Length of a spiral (parametric)

  1. 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. jcsd
  3. dextercioby

    dextercioby 12,328
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    Do you know how to calculate the length of curves?In general,what formula need you apply...?

  4. 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.....
    Last edited: Feb 2, 2005
  5. dextercioby

    dextercioby 12,328
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    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...

  6. 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....
  7. dextercioby

    dextercioby 12,328
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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.

  8. 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....

  9. dextercioby

    dextercioby 12,328
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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...?

  10. man this is embaressing but I remember so little of this stuff

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


    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....
  11. dextercioby

    dextercioby 12,328
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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.

    Last edited: Feb 2, 2005
  12. Determining X and Y coordinates along a spiral

    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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