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While v-carving with a CNC router, the width of the cut is determined by depth of the Vee bit in the material. Simply stated, the deeper the bit goes, the wider the carving. I've successfully calculated the corresponding/matching depth if using a ball nose bit (half sphere).

I've been studying ellipse formulae but can't come up with means to calculate the matching depth if using a different type of bit called a round-over bit. Its profile is vertically elliptical.

Let's assume the major axis is $0.125$ units with a minor axis of $0.1133$ units. Some bits have a small offset that needs to be accounted for as well. Note the "C" in the diagram.

The known value is "A", what I need to be able to calculate is the "B"; the round-over bits corresponding depth to attain the same width.

I understand there are small changes in the thousandths for differing heights but that is critical in Vee Carving text or images. I'm currently importing the original G-Code that drives the machine into a spreadsheet that I can crank out revised numbers for ball nosed bits. All I need now is the formula that does it for the Round-over bits. Been struggling for quite some time to figure this one out.

Any help would be appreciated.

Diagram

Example Available Bit

Sarvesh Ravichandran Iyer
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StevenM
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  • Welcome to Math.SE! <> To clarify: (i) Are the ellipse cross sections vertical at the bottom of the bit (where the offset C is labeled)? (ii) Is the problem to find the depth so that the round-over bit cuts the same width at the wood surface as a Vee bit at the indicated depth? – Andrew D. Hwang May 19 '22 at 16:56
  • Elliptical equations is something entirely different. You are looking for assistance in applying the conic equation of an ellipse to a practical woodworking problem. I have eliminated the elliptical equations tag and added the applications tag to improve the chances of the right people looking at your problem. – John Wayland Bales May 19 '22 at 17:46
  • @AndrewD.Hwang (i) Yes. (ii)That is precisely what I'm trying to accomplish – StevenM May 19 '22 at 18:27
  • @John Wayland Bales Thank you for the changes. As you can see, I'm a newbie here and thought I had that appropriate tag. – StevenM May 19 '22 at 18:32

1 Answers1

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Referring to the diagram, let $a = 0.1133$ and $b = 0.125$ denote the semi-minor and semi-major axes of the ellipse (really, the half-width and half-height, regardless of which is larger), and let $w$ denote the width of the cut at the wood surface. The depth $B$ above the bottom of the bit is given by the formula $$ B = \frac{b}{2a}\sqrt{[4a - (w - c)](w - c)}. $$

Cross section of a round-over bit

To see why, put coordinates at the center of the right ellipse. If $x = a + \frac{1}{2}(c - w)$ and $y = B$ denote horizontal and vertical positions with respect to the center, the ellipse has equation $$ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. $$ Substituting the expressions above for $x$ and $y$ gives $$ \frac{B^{2}}{b^{2}} = 1 - \biggl[1 + \frac{c - w}{2a}\biggr]^{2} = \biggl[2 + \frac{c - w}{2a}\biggr] \biggl[\frac{w - c}{2a}\biggr] = \frac{[4a - (w - c)](w - c)}{(2a)^{2}}, $$ and solving for $B$ gives the formula above.

Andrew D. Hwang
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