Spreadsheet for Determining Radius of a Curve

This post will be considered by some as an intellectual exercise only.  Maybe it's for math geeks, like myself.  There are simpler methods to do what I write about.  I'm all for simplicity, but I like the math.

When I want to put a curve on the underside of a chair rail or a table apron, a simple method is to place a clamp at either end of the intended curvature and bend a stick (or ruler) to the desired "bulge" of the curve.  While this method will give attractive results, the resulting curve will not be exactly circular.  The stick bends more at it's center than at its ends.  And it may not bend equally both sides of center.

Marked lines 1" from ends, and placed clamps near those marks

Centerline marked and 1 inch "bulge" marked from lower edge

Bend a stick to the "bulge" mark, and draw the curve

Here's the resulting curve

There are times when I want to end up with a curve that is part of a circle.  And when this is the case, I'd like to know the radius of the circle that will give the desired curve so that I can lay it out on the workpiece.

Here's an example, same as in the above pics.  Suppose I have a 20" x 2 1/2" rail and I want to put a circular arc on the underside.  I want the arc to start 1" from each end of the rail and I want it to extend up into the rail's front face by 1".  So that's an arc 18" wide with a 1" bulge.

Using a little math, I can calculate that the radius of the circle that gives the appropriate arc is 41".  I can then use a stick (or piece of string) of that length and a pencil to lay out the curve.  I'll write more about the math below.

This stick will allow arcs up to approx. 48" radius

One end has a shallow slot to run a pencil in

You can put a screw anywhere on the stick for whatever radius you need.
The screw tip exits the bottom side, and it is used as a pivot point for drawing the arc.

Here, I'm using the radius jig to mark the arc on the workpiece.
Note that the screw has to be in line with the centerline of the workpiece.

The two methods give very similar results, so it really doesn't matter which you use.

You can see the slight difference in the two methods

But if you're doing a smaller piece, say an arc only 6" wide, then that bendy stick won't bend in that tight a space.  That's when it would be easier to use a radius stick.  For those smaller pieces, I can use my homemade trammel points to create the arc.  I'll show that below.

Another place where drawing a circular arc has come up is when laying out a camber on a scrub plane iron.  I recently made an iron that was 1 1/2" wide and I wanted a 1/16" bulge at the cutting edge.  For this iron, I simply filed a curve to get it close.  But if I wanted to be more precise, I could figure out what the radius of that curvature is and make a template.  Turns out it is about 4 1/2".

Cardboard template, 1 1/2" wide with centerline drawn

Homemade trammel points

Setting the distance to 4 1/2"

Drawing the appropriate arc on the template

Measuring the resulting bulge - it's about 1/16"

Comparing the template to the plane iron

OK, now here's the math part.  It has to do with right triangles and the Pythagorean theorem.  I'll start with the example of a table apron, which I want to put an arc on the bottom edge, starting some distance in from each end and rising up a certain amount.  In the picture below, points A and B are the ends of the arc, and C is a point centered between them.  The width of the arc I'll call "w", so the distance from C to B is 1/2 w.  The bulge of the arc is the distance from the lower edge of the board to the highest point on the arc.  I call it "b".

The workpiece with arc drawn.  The arc has width "w" and height "b".

Now I'm going to zoom out so you can see the rest of the picture.  The arc drawn on the workpiece is part of a large circle, whose center is at point O.  The radius of the circle, given by distance R, is the distance from point O to any point on the circle.  I'm using point B here.

Zoomed out pic shows the complete circle with radius drawn

Another radius is from the center, O, to the point at the top center of the arc, drawn straight up from the center.  That radius is made up of two distances, the distance from O to C, and the distance from C to the top of the arc.  The latter of those I've already called "b", the bulge.  So the distance from O to C is a radius minus the bulge, or R-b.  Triangle OCB is a right triangle, so from Pythagoras, the sum of the squares of the two legs (legs are the sides of the triangle making the right angle) equals the square of the hypotenuse.

R^2 = (R-b)^2 + (w/2)^2       (the symbol ^2 means squared)

R^2 = R^2 - 2Rb + b^2 + (w^2)/4

And simplifying, we get:

2Rb = b^2 + (w^2)/4, and then

R = (b^2 + (w^2)/4) / (2b)

So to draw an arc with a certain width and bulge, you can calculate the radius of the circle that describes the arc and use a stick to draw the arc.  I've put this formula into an Excel spreadsheet.  If you want to do the same, type the following in an Excel sheet:

=(C4^2 + (C3^2)/4)/(2*C4)

In this formula, C3 is the location in the spreadsheet where I entered the arc width (not half width, the whole width).  C4 is the location in the spreadsheet where I entered the bulge.

Here's a picture of what my Excel sheet looks like (replace the word "Iron" with "Arc")

The example shown above has an arc width of 18" and a bulge of 1".  The resulting radius is 41".

Here are the details for the other example of this - shaping a plane iron with camber to make a scrub plane.  It works exactly the same way.

The plane iron drawn in gray, with markings like the earlier example.

Here's the upper end shown closer up

Using the spreadsheet for an iron that is 1 1/2" wide and giving a 1/16" bulge, I get a radius of 4 1/2".

The cambered iron example

For anyone who is actually interested in this stuff, if you can't figure out how to get the formula I wrote above (shaded in yellow) to work in your own Excel spreadsheet, contact me using the "contact me" gadget somewhere on this blog page.  Specify that you want the radius spreadsheet, because I've got another spreadsheet for a different application and don't want to send the wrong one.