Morris Dovey’s Cove Cut Spacing

[Cutting a cove]
Cutting this cove in a 2x4 with a 1-inch round nose bit was easy.

The lower (black) arc represents the desired cove profile; and the upper (red) arc represents the position of the center of the (hemispherical) cutter. The two circles represent two of the possible positions of the cutter while cutting the arc profile. The shaded area shows the ridge that would remain after cutting.

You can see that if the cutting passes are closer together, the ridge will be less pronounced. What I want to do is to choose the height of the ridge and use that height to determine the spacing between cuts.

With that in mind, let's remove the cutter circles and add our chosen ridge height (the short vertical blue line) to the drawing. At the top of the ridge line, we'll add a circle (blue) with the same radius as the cutter we're going to use.

This is the locus of all points at a distance r (the radius of the cutter) from the top of the ridge.

At the two points where this circular locus intersects the toolpath (red) arc, we have the positions of the center of the cutter when such a ridge is left.

Here I've redrawn that circular locus in grey; and the two cutter circles in red. This shows how we can use the ridge height and the cutter radius to locate two adjacent cutter positions.

What we're interested in is the angular distance between the centers of the two cutter circles; so let's redraw.

Let's do some simple algebra. Our variables are:

R = radius of our arc profile
t = radius of the tool position locus
r = radius of the cutter
e = height of the ridge above the ideal profile
A = angle between two adjacent cuts

Let's use Cartesian coordinates and let the origin be at the midpoint of the arc profile.

We know that:

The center of the profile arc and the toolpath locus is at (0,R).

The center of the locus of all points equidistant from the peak of the ridge is at (0,e)

The equation of a circle of radius r with center at (h,k) is:

(x - h)² + (y - k)² = r²

Then the equation of the ridge locus is:
     x² + (y - e)² = r²      [1]

and the equation of the toolpath locus is:
     x² + (y - R)² = t² = (R - r)²      [2]

Subtracting [1] from [2] yields:
     (y - R)² - (y - e)² = (R - r)² - r²      [3]

Which can be expanded and then simplified to:
     y = (2Rr - e²)/(2R - 2e)      [4]

We can substitute this value of y from [4] into [1], expand, and determine that the values for x are:
     x = ± sqrt(r² - (y - e)²)

And so, as expected, we have intersections on both sides of the origin.

If we define a to be one half of the angle A, we can use the right intersection point to determine:

     tan(a) = x/(R - y)

which we can write as:

     a = arctan(x / (R - y))

and to get the entire angle:

     A = 2a = 2arctan(x / (R - y))

Note for new CNC woodworkers:

I normally use the largest round nose or core box router bit (typically 1" diameter) that I have available and set e = 0.001" for my production work. A 1/1000" ridge is, for all practical purposes, undetectable.

Copyright © 2004 Morris R Dovey