What is difference between Archimedian spiral and involute of a circle?

From Wolfram MathWorld article on Circle Involute Pedal Curve:

The pedal curve of circle involute

f = cos t + t sin t
g = sin t – t cos t

with the center as the pedal point is the Archimedes’ spiral

x = t sin t
y = -t cos t.

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@prasad, it looks like you start with a circle, then take its involute, which is approximately an Archimedes’ spiral, then take the pedal curve of the involute, which is exactly an Archimedes spiral.

The two curves are compared in this image from Wikipedia (black = circle involute; red = A. spiral).

Wow, @gasman I was waiting to see who would answer this question!

@prasad I just realized I posted the very same image you put in your question – guess I’ve been staring at spirals for too long…

Thx @Earthgirl – Hardly off the top of my head – I had to read up on it.

@gasman Your links and explanation clears my reservations about the pedal curve (and I got to know of the pedal point too). After reading through the links and some other material, I infer that Archimedes’ spiral starts from origin while the involute starts off the origin (on a circle centered at origin). So it is near the origin that Archimedes’ spiral is not a parallel curve; the distance between coils of the Archimedes’ spiral is not constant near the origin, for subsequent coils it is a parallel curve. Involute of a circle is a parallel curve throughout. Please correct me if I am wrong.

@Earthgirl I wondered too and was not sure if I should ask this question on fluther. But then, I relied on my experience that there are some intelligent jellies here, and it is once again confirmed!

I have found some other definitions that explain both the curves in other words.

The involute is best visualized as the path traced out by the end of a piece of cotton when the cotton is unrolled from its reel.

The Archimedean spiral is the locus of a point that moves away from another fixed point at uniform linear velocity and uniform angular velocity.
It may also be considered to be the locus of a point moving at constant speed along a line when the line rotates about a fixed point at constant speed.

@prasad Is this strictly theoretical to you or is there some application for this in your engineering work?

@prasad Parallel curves are usually a family of separate curves, while the spiral is sort of its own parallel curve. The distance between coils equals the distance between points exactly one turn (2π radians) apart. If the two points are (x1,y1) at t and (x2,y2) at (t+2π)

then
x2 – x1 = (t + 2π) sin (t+2π) – t sin t
= 2π sin t
and
y2 – y1 =(-t – 2π) cos (t+2π) + t cos t
= 2π cos t

The distance between points on successive turns √[ (x2-x1)^2 + (y2-y1)^2 ]
= 2π √ (sin^2 t + cos^2 t)
= 2π.

Btw the groove in a phonograph record is a nice model of an Archimedean spiral.

By a similar analysis of the parametric equations you can verify that the distance between successive turns of the circle involute is also exactly 2π. The two curves are displaced less than that from each other, however, as seen in the graph you posted.

@Earthgirl A little on theoretical side. I am to teach how to draw these curves, but I thought I should better know more about it. When I read about how to draw both these curves using a string, as shown in involute and the Archimedes’ spiral, then I was confused; both methods look quite similar.

@gasman I must say your explanation facilitates me very much to understand different concepts clearly. What I read about above things from the above linked pages, it was not comprehensible for me until I read your posts. I will remember the example of a groove in a phonograph record; and some times such examples help me understand better!