Calculating Radius of a Cambered Iron

So you want to add a camber to a plane iron and don't happen to have a round object with the curvature you're looking for.  In this post, I'll use geometry to figure out the radius so you can draw the arc with a compass.  I know, for some people I'm overcomplicating this, but for this analytically minded woodworker, it works.

This came up for me last week when I did some maintenance on a wooden jack plane.  After sharpening many, many times, the curvature of the iron wasn't as it should have been, so I set it right.  I needed to figure out the radius of curvature that would give me a 1/16" bulge on an iron that was 2 3/16" wide.  By the "bulge", I mean the amount the camber extends further than the corners of the iron.

For this blog post, I'll simplify the math by using a 2" iron width.

Artist's rendition of a 2" wide plane iron with 1/16" bulge

Closeup of the business end

I've labeled two points on this picture: Y is the right corner of the iron and Z is the point at the center of the iron directly over from Y.  The "bulge" is 1/16", the amount past Z that the iron extends.

In the next pic, I've drawn a circle whose curvature matches the camber of the iron.  The point labelled X is the center of the circle.  Also drawn is a line from X to Y, and since Y is on the circle, segment XY is a radius of the circle.

Circle drawn that matches curvature of the camber

Zooming in again, we see a triangle formed by XYZ.

Triangle XYZ

We have enough information now to calculate the radius of the circle, and from that, we can draw a template of the camber to be transferred to the iron to guide grinding and sharpening.

Triangle XYZ is a right triangle.  The Pythagorean theorem tells us that YZ^2 + ZX^2 = XY^2.  (Read "^2" as "squared".)

XY is the radius of the circle: I'll call it R.

YZ is half the width of the iron, in this example 1".

ZX is slightly less than a radius, by 1/16", so ZX = R-(1/16).

So we get:   1^2 + (R - 1/16)^2 = R^2

Simplifying: 1 + R^2 - (1/16) R - (1/16) R + (1/16)^2 = R^2

Simplifying: 1 -(1/8) R + (1/256) = 0

Simplifying: 1+(1/256) = (1/8) R

And finally: R = 8 * (1 + 1/256) = 8  1/32"

In the case of my jack plane, the iron is 2 3/16" wide and the calculation came out to give a radius R =  9.6" ( I rounded to 9 5/8").

If one wanted to extend this to any bulge (call it B) and any width iron (call it W), then the formula comes down to:

R = ((W/2)^2 + B^2)/2B

I know this is not for everybody.  In fact, it's probably a lot easier to simply draw an arc using a nail, a string and a pencil, changing the string length until you get the right bulge on the arc.  But I like the analytical approach.  Math has served me so well over the years.