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Involute of a Circle

Date: 01/22/99 at 10:44:30
Subject: Formula for the involute of a circle

I know how to construct the involute of a circle, but I can't find a 
formula for it.

Date: 01/22/99 at 11:53:45
From: Doctor Anthony
Subject: Re: Formula for the involute of a circle

A string is wound around a circle and then unwound while being held 
taut. The curve traced out by the point P at the end of the string is 
called the involute of the circle. If the circle has radius r, the 
circle centre is at the origin O, the string is tangent to the circle 
at Q, and the initial position of P is (r,0), then the parameter, 
theta, is chosen as angle between a radius line to Q and the x-axis. 
The parametric equations of the involute are: 

   x = r[cos(theta) + (theta)sin(theta)]
   y = r[sin(theta) - (theta)cos(theta)]

- Doctor Anthony, The Math Forum   

Date: 08/08/2001 at 11:19:37
From: Scott McConnehey
Subject: Involute Form on Splines and Gears

Dear Dr. Math:  

I'm not a mathematician but normally I can use existing formulas and
equations effectively. I saw your formula for the involute of a circle
and tried to plug in some values but came up with what I viewed as
unrealistic numbers. In the formula X=r(cos(theta)+(theta)sin(theta)), 
does (theta)sin(theta) mean angle times the sin of the angle? For 
instance, if theta is 45 degrees, do I figure this as 45 degrees time 
the sin of 45 degrees? 

When I take this approach, I seem to get unrealistic numbers. In this
case, I know that the value for "X" can't be 32.5269 using a 1-inch
radius. Could you please tell me what I'm doing wrong or point me in the
right direction if there's a resource I can access that would make this a
little clearer for me?  

Let me explain that I work in a navy calibration lab and I need to 
derive a method for verifying the involute form on spline teeth.  
Anything you can do to help would be greatly appreciated.


Date: 08/08/2001 at 12:36:29
From: Doctor Rick
Subject: Re: Involute Form on Splines and Gears

Hi, Scott.

The angle theta in Dr. Anthony's response is assumed to be in radians.
The formula is derived in part from the length of the arc subtended 
by a central angle theta on a circle of radius r. This arc length is
r*theta, but only if the angle is in radians. There are 2*pi radians in a
full circle (360 degrees), so that the "arc length" of a full circle is
2*pi*r, the circumference of the circle. 

To convert degrees to radians, multiply by pi/180. Thus, 45 degrees

  45 * 3.1416 / 180 = 0.7854 radian

- Doctor Rick, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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