Length of a spiral (parametric)

[Mounty]

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.

 

[dextercioby]

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

Daniel.

 

[Mounty]

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]

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]

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]

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 want to 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]

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]

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]

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]

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.

 

[E=MC**2]

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?