# What are some tricks to draw long arcs along a table apron

I'd like to cut an arc along a table's apron, like in the image below. To scribe an arc circle for a gentle curve, e.g. 1" in height at the apogee and spanning a 48"-long apron, one would need to scribe an arc on a circle with a radius of 24 feet. This quickly becomes impractical.

(The forum where that image is from suggests a few things, like bending sticks, but maybe there are other easier solutions)

• A bent stick (bent lath in older terminology) is still about the best way to do this in terms of simplicity and directness of approach. For best results you have to select the lath carefully for straight grain and no knots, which can be easier said than done. Of course these days with cheaper long steel rules available those can often be used instead and virtually guarantee an even arc (AKA fair curve) up to the usual practical limit of 3' or 1m. – Graphus Jun 19 '18 at 13:46

I've collected a few solutions online, and a website to help with some of the steps:

• Pin a bent stick (or lath) on three points (e.g. center and extremities), and draw its outline. It might not be perfectly circular, but you'll be able to adjust the curve. For best results, you would have to select the stick/lath carefully for straight grain and no knots. You could also try bending a drywall T-Square on its side, depending on the length and curvature of the arc needed.

• Print the curve on tiled printer paper, transfer the pattern to cardboard or thin rigid stock to make a stencil, and then use the stencil to scribe the line. There is a website called arceng which can generate various types of (printable) curves.

• Pin one end of a string or of a rigid board in the center of the circle. Tie a pencil or a marker to the other end, make sure the string is taut (pulled tight), and revolve around the center pin to draw the arc. That won't work for large radii, naturally.
• Calculate the location of a discrete set of points along the curve to make a spline, and link them with short smooth segments. Again arceng can provide those points for simple curve functions, namely circular arcs.

I'm sure there are more generic procedures and tools to draw arbitrarily shaped outlines (e.g. the outline of a carved Queen Anne headboard), but I don't know them. If you have a plan for an existing curve design at a different scale, you would probably just enlarge it on paper and transfer it to a rigid stencil.

If you intend to draw a lot of curves, a tool called a "ship curve" or "blending curve" might turn handy, but they can be pricy. A plastic ruler on its side might offer a good-enough substitute, in a pinch.

• arceng also supports metric measurements. Using mm would be easier than using fractional inches – Eli Iser Jun 19 '18 at 10:03

Here's a neat method I found on woodgears.ca:

Let A and B be the endpoints of your circular arc, and let M be the midpoint between A and B. If P is the highest point on your arc (the apogee), then let h be the distance from M to P. Finally, let L be the distance between A and B: Note: Considering the triangle formed by the center point, point M, and point A (or B), you have, from the Pythagorean theorem, that r2 = (L/2)2 + (r-h)2. Solving for r, you get r = (L2 + 4h2)/(8h), which could be too large for a typical drawing method (long string with a pencil, or a beam compass).

Rather than working with a super long radius, you can use a fact of geometry: the inscribed angle formed between two points of a circle (A, and, B) and any other point along the perimeter, is constant. Set two nails at points A and B, and get two straight sticks, each a bit longer than L, butted against each other at an angle.

You can use the following image as a guide. Rest the sticks (depicted by the pairs of colorful lines) on the arc-side of the work piece, so each stick touches a nail, and cross the inside edge of the sticks at point P. At that point, the yellow lines are your sticks, the bright green point at their intersection is where you hold the pencil.

Now, holding a pencil at the intersection of the sticks, and keeping the angle the same between the sticks, slide the connected sticks left and right until the pencil has reached both nails. For instance, to draw the right half of the arc, you would move from the yellow position, to the orange position, then the dark orange, etc., until you reach B. You will have drawn a circular arc between A and B with required height h. This will work as long as the angle between the two sticks is kept constant (see theory here). You would record the angle at point P (because you can precisely calculate that point's position), and then if you keep the same angle when sliding, you will "reveal" your other arc points. You could glue the sticks at the angle you want, or use a protractor to ensure the angle is kept constant.

• Geometrically speaking, an excellent answer. But note that the fixed sticks will, for the 4-foot curve specified, be 8 feet long. You could very usefully do this by cutting a strip from a full sheet of plywood, then cutting the appropriate angle using a 4-foot straightedge, but this is going to be pretty awkward to use. – WhatRoughBeast Feb 20 '19 at 4:32
• @WhatRoughBeast The sticks would each need to be 4 feet long for that curve (not 8). – user138719 Feb 20 '19 at 5:12
• Yes. And they are fixed together essentially end-to-end, and the assembly will be 8 feet long. – WhatRoughBeast Feb 20 '19 at 6:14
• Ah yes -- I misunderstood you. If 8 feet seems too big, I suppose one of the sticks could be just a bit longer than 2 feet, and you could use it to draw half the arc with the long side touching the nails. Then flip it over to draw the other half arc. – user138719 Feb 20 '19 at 15:26