The Reverse Curve
by Jon Siegel
This article first appeared in The Old Saw, the journal of the Guild of New Hampshire Woodworkers (www.gnhw.org) February 2006.
In recent years, I have taken up playing pocket billiards, commonly known
as pool. There are many reasons for this infatuation. For one, I am trying
in some futile way to recapture my misspent youth, as I remember how much
fun I had playing pool years ago. Pool is such a beautiful game which reveals
many aspects of geometry and physics in a simple and elegant way. My enjoyment
of the game is enhanced by playing with pool cues I made myself, and turning
pool cues is possibly the only thing (or two) I can think of that is more
fun than playing pool.
See What You Are Looking At
But a further reason is that I think my woodworking skills can be improved
by playing pool, because pool teaches me to SEE WHAT I AM LOOKING AT. Years
ago an old timer said to me, “To be a cabinetmaker, you got to have
good eyes.” Back then I didn’t understand what he meant. But
now, decades later, I think I do. You have to learn to SEE things because
woodworking, like all sculptural arts, is intended to create a three dimensional
visual impression.
A simple example of this occurs when we visit the lumber yard and sort through
the boards in the rack. We are looking for warp, or other defects in the
shape of a piece of lumber. We sight down the board for crook or twist. This
is an acquired skill, but with some practice one can do it in seconds without
thinking very much.
Seeing a bump in a straight line is one thing, but in this article I will
teach you how to visualize complex curves. With this information you can
design shapes which present the visual impression you want to project, locate
errors in curves, and produce the curve at the lathe using only your eyes
and a single caliper measurement as a guide.
The Vocabulary of Shapes
Spindle turnings are composed like music. The various elements are arranged
in sequence along a line. Each one leads into the next, and must harmonize
with its neighbors and to the whole form. None of the elements mean very
much on their own, just as a single note played on a piano doesn’t
make much of an impression. It is only the way they relate to each other
that gives the turning elegance, grace, a sense of proportion, and finally
the aforementioned sculptural visual impression we are seeking.
There are a small number of shapes which make up the vocabulary of the woodturner,
but of course their variations and combinations are infinite. These are straight
lines, simple curves and complex curves. Simple curves are either convex
(beads and ball forms) or concave, (coves and hollows). But by far the most
powerful shape used in woodturning is the reverse curve, or S-curve, because
it is a combination of both convex and concave. This is sometimes referred
to as the “vase form”, but it should be noted that it is equally
effective upside down or on horizontal members such as stretchers.
Elements of the Reverse Curve
Figure 1 illustrates all five elements of the reverse curve:
1 Overall length
2 Location of the high point
3 Location of the low point
4 Location of the inflection point
5 End point angles
The overall length is labeled (L). It is measured parallel to the axis (axial
length) and is not the length of the curved line itself which of course is
slightly longer. When commencing to produce the curve at the lathe, the overall
length is the first item to locate and mark out.
The high point is labeled in the figure. It is the distance from the end
of the curve which is critical, that distance is labeled (A). Nearly always,
the high point will have a diameter which is full, that is, the maximum diameter
of the workpiece. It is therefore easy to mark this point with a pencil line,
and leave it there until the shape is nearly completed. But do not leave
a flat spot at the top of the curve. This is a very common error, and it
should be noted that sanding worsens this problem, because the action of
the sandpaper is more aggressive on the flat grain, right at the top of the
curve. So, if anything, it is best to leave a little bump at the high point,
since it is easy to fix this in the sanding stage, whereas a flat spot at
the high point is almost impossible to fix by sanding.
The low point is labeled, and again, it is the distance from the end that
is noted. This distance is labeled (B). The low point has two elements which
need to be controlled simultaneously – the distance from the end (B),
and the diameter. Here is where a caliper is essential for setting the most
important measurement – the diameter at the low
point. Some turners
make a parting tool cut straight-away at the low point. I have never found
this method to be satisfactory, because for one, it weakens the turning at
an early stage which inhibits the roughing out process, and secondly it interrupts
the movement of the chisel over the work at the middle of the curve. In other
words, the chisel crashes whenever it crosses over the parting tool cut,
making it difficult to maintain a smooth line as you approach the finished
surface.
Woodturning’s Best Kept Secret
Here is something you won’t find in any woodturning books. But it is
the MOST IMPORTANT element in helping you visualize the reverse curve. I
already mentioned that the reverse curve combines both convex and concave
parts. Some people mistakenly believe there is a straight section between
these two elements. But no part of the curve is straight. The convex and
concave portions meet at a place called the INFLECTION POINT. Its location
is shown with regard to its distance from each end of the curve – the
distance from the top (C) and the distance from the bottom (D). In this example,
the location of the inflection point is above the center of the curve, and
(C) and (D) have a ratio of about 2:3. This has the visual effect of making
the form fuller and more robust than a curve with the inflection point in
the middle. 
To further illustrate this point (pun intended), Figure 2 show three versions
of a turning with a reverse curve. The difference is the location of the
inflection point. Notice that these turnings do not look at all alike even
though the length and diameter measurements are the same. This shows how
important it is to pay attention to the location of the inflection point
in designing and producing reverse curves.
Finally, the remaining element which defines the curve is the END POINT ANGLE.
Every curve has two end points. A line tangent to the curve at the end point
forms an angle with the radial line and these are shown as P° and Q° in
Figure 1. End point angles are critical to simple curves such as beads and
coves as well as complex curves such as the reverse curve shown here. 
The most common design error is end point angles that are too large. This
results in features that are shallow, poorly defined, and do not make sharp
lines at their end points – Figure 3. These sharp lines are formed
at the intersection of the surfaces and are critical to making the details
(with light and shadow) which set off the large features. As I will discuss
further in future articles on design, it is the balance of the details to
the large features which constitutes the most critical factor in good design.
Therefore, curves should terminate in sharp lines, and these lines must not
be blunted by careless sanding. The intersections do not have to be 90 degrees,
but they should be near 90 most of the time. Therefore end point angles should
rarely be more than 25 degrees.
Even something which seems complex, can be easily visualized by breaking
it down into its individual elements. I hope this article has helped you
understand what those elements are and how to see what you are looking at.
Nine ball in the corner pocket!