Morris Dovey’s CNC Cove Radius Calculation

Calculating the radius of a cove h units deep and w units wide

The equation of the line joining the midpoint of the arc to the right endpoint of that arc is given by:

     y = ((h / 2) / (w / 4)) * x

     y = ((4 * h) / (2 * w)) * x

     y = (2 * h / w) * x

The slope of the line from the midpoint of the chord to the arc's center point is:

     m = -w / (2 * h)

And so the equation of the line connecting the midpoint of the chord to the arc's center is:

     y = m * x + r; where r is the y-intercept (and the arc's radius!)

Substituting at (w / 4, h / 2):

     (h / 2) = (-w / (2 * h)) * (w / 4) + r

     r = (h / 2) + (w / (2 * h)) * (w / 4)

     r = (h / 2) + (w * w) / (8 * h)


Determining the angular width of the cove

We can see that:

     tan A = (w / 2) / (r - h) = (w / (2 * (r - h)))

So one-half of the angular width is given by

     A = arctan(w / (2 * (r - h)))

And the entire angular width is simply 2 * A

Copyright © 2003 Morris R Dovey

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