Nice Curves, No Math | Popular Woodworking Magazine

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Mechanical solutions for the demanding carpenter. was a time when I was pretty good at math, but that was when “personal” and “computer” were two words you never expected to see together. A few decades later, I often find it difficult to run multiple tasks long enough to add two numbers together, also remembering why I added them in the first place.

So while I’m still capable of doing most of the shop math – at least briefly, after my sixth cup of coffee – I tend to opt for mechanical and visual solutions when they are viable.

Fortunately for us confused-headed guys, there are methods of drawing curves that will more than adequately approximate the beautiful shapes generated by higher dimensional (admittedly fantastic) parabolic equations, as demonstrated by Bruce Winterbon in his “Bow Shelves” article.

Here are three that I used in my shop.

Arch method

This method (shown in the opening photo) is probably the most commonly used technique for drawing clear curves on furniture. It’s definitely the one I use most often. The basics are simple: First, find a sturdy shape that bends elastically (that is, comes back when released). Fortunately for carpenters, thin strips of wood fill the bill nicely.

I use it more often 18“X 34“Straight grain hardwood strips, anywhere from 18” to 48 “in length. At both ends, I cut a small slit to fit a piece of string or twine and knot the ends of the string to maintain the curve I want.

Mathematically, these curves (called “elastic”) aren’t simple at all, but Mother Nature does all the math and the resulting curves are visually pleasing.

Keep in mind that variations in wood thickness alter the shape, which can be very useful for varying curves. By selectively thinning the wooden arch it is easy to vary the rate of curvature quite significantly. The thinner an arc area, the more sharply it curves. Thin the arc from one end to the other and you will shift the “peak” of the curve to achieve a beautifully faired non-symmetrical shape.

A few minutes of experimentation with a handplane is all it takes to get hooked on these devices and their flexibility (pun intended a little).

To best approximate the curve given by the “hyperparabolic” calculations to create the “bow brackets”, the bow must be tapered at both ends, increasing the degree of curvature at the ends.

Jump rope method

Strings attached. A jump rope or thread can be hung at two points to generate a catenary.

I don’t find this method as comfortable as arching because it can only be done vertically, but it’s simple to do. Just hang a piece of string or twine between two posts and let it hang down. This is the shape you see in suspension bridges, skipping ropes, and Christmas tree lights hanging from eaves. Although the shape looks a lot like a parabola (which you may remember from your geometry lesson), it is technically known as a “catenary” and is a more complicated mechanical function. However, visually it’s a minimal difference, and this is an excellent and easy way to generate a nice curve.

There are two things to be aware of when creating a catenary.

Note that the pins used here are outside my working curve.

First, the string used must be as close as possible to the perfect flexibility you can find. Strings with any “memory” (meaning they hold a curve) will not lock well. The two materials that I have been most successful with are (of course) a good quality jump rope and simple sewing thread. Both will give you pretty good turns.

Second, there is inevitably some distortion of the curve near where it joins the pole. This problem can be solved more easily by making sure that the poles themselves are outside the corner area.

As you can see in the photos on the previous page, the wire catenary I made is actually a pretty good approximation of the hyperparabolic curve. It’s not perfect, but it’s close enough to be happy enough with the results. This is probably the method I would use for this particular project.

Cone of light method

Conical sections. From a cone of light you can create (descendant) an ellipse, a circle, a parabola and a hyperbola.

The four curves that we probably see most often in woodworking – the ellipse, the circle, the parabola, and the hyperbola – are collectively known as the “conical sections”. They earn this name because all four can be made from the intersection of a plane and a cone. Imagine, if you like, a samurai slicing a birthday hat (maybe he hates getting old – I don’t know). If you cut the hat generally horizontally, the opening at the top will be an ellipse, or the special case known as a circle if the cut is exactly parallel to the base of the hat.

If the samurai cuts the hat in a generally vertical direction, however, it will produce a hyperbola or parabola (unless it splits sharply across the dot, but we will ignore this possibility). A cut exactly perpendicular to the base of the hat will result in hyperbola. A vertical cut out will produce a parabola.

None of this is particularly important, unless you’re at a birthday party with the samurai, but it does allow for a very easy way to reproduce these curves in the shop. Instead of stocking up on a stash of different sized birthday hats though, just use a beam of light to create the cone. And instead of cutting it with a sword (because you can’t, not even with a katana), use a comfortable flat surface like a plane.

Just take a flashlight (or any light source with a circular screen) and project it onto a wall. If you project it more or less straight onto a wall, the light beam will form an ellipse. If you point it exactly perpendicular to the wall, the ellipse will be a perfect circle.

Now aim the light closer to parallel to the wall. When the beam looks out of the wall, it will generate a parabola. If you place the light very close and pointed exactly parallel to the wall, it will generate hyperbola.

Note that this method has a significant drawback – it is difficult to hold a torch with precise orientation and draw the resulting curve at the same time. When I used this method, I do it with a flashlight mounted on a photo tripod. If you don’t have a tripod, you can use an inexpensive clamp shop lamp instead of a flashlight. Just make sure it has a round frame.

One last tip before you go out and generate a million curves: although all of these methods theoretically generate perfect shapes, in the real world this is almost never the case. There are small variations in materials, positioning etc.

As a general rule, whenever I make a symmetrical curve of any kind, I only copy half of the curve of the wood, catenary or light image. Then I copy that half of the curve to create another perfectly symmetrical half.

To see? Nothing to fear here. Apart from the samurai parties, anyway.

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