On Fri, 13 Feb 2004 21:52:07 GMT, Bay Area Dave wrote:

HOw can I create a fair curve about 36" long that isn't merely
an arc of a circle.

Some more comments on smooth curves and French curves.

A "smooth" curve is one that avoids corners. A "corner" is a point
where there's a discontinuity in the tangent - i.e. a tangent drawn
just a tiny fraction to one side of the corner wouldn't line up with a
similar tangent drawn to the curve just the other side of the corner.
By implication, if we can align the tangents of two curves, then we
can avoid the corner.

Now the fun bit - this even works with two different curves. If we can
just get those tangents to line up, then the transition between them
will be smooth - no matter how much of a change in curvature is
involved.

Aligning tangents is one approach. But if tangents align, then so must
the perpendiculars align. Now for a simple arc, the perpendicular is a
radius, which suggests a useful special case.

When joining a circular arc onto a straight line, the centre used by
the compasses to strike the arc must be on a line perpendicular to the
straight line, and crossing it at the point where the curves join.
This sounds awkward, but it's pretty obvious in pictures. It's also
the principle behind constructing the range of Gothic arches:
http://codesmiths.com/shed/workshop/techniques/arches.htm


A French curve is a curved stencil whose shape is a mathematically
somewhat complex curve that changes its radius of curvature along its
length. There are several sorts, and those with access to Mathematica
http://mathworld.wolfram.com/FrenchCurve.html
or other drawing tools can probably generate their own. A useful
source for some basic curves of this family are the Archimedian and
logarithmic spirals, which can be constructed with simple paper and
pencil methods.

A French curve is a curve that changes along its length. So you should
be able to roughly eyeball what you need, then find a curve you
already have that offers roughly the right curvature at two places
roughly the right distance apart (the previous trick about making
enlarged MDF duplicates by photocopying is worth it, if you need big
furnituremaker's curves.)

To use a French curve, first choose the best fit you have. The place
it roughly in place so that it touches the existing curve at the right
places. Now slide it from side to side, keeping it aligned against
those end points, until the tangents are a good fit. You're done !

If it doesn't work, then try different curves. They'll all fit and
give a smooth curve, but they might not offer quite the radius you're
after. _Don't_ use French curve by picking the radii you want,
plonking it down and then trying to "smooth out" the corners. They
just don't work that way. Each given curve will only fit at one place
- if you want different combinations of radii, then you need a
different curve (but photocopiers with fine enlargement control are
your friend).

Avoid Bezier curves (offered by any CAD program) for freehand drawing
work. Although they can be used, it's all too easy to make a spline
curve look ugly in a purely aesthetic sense (and the maths of exactly
_why_ is both horrible, and does actually exist). However today is
Valentine's Day, and I have taken a solemn oath to forswear all
aesthetics.



--
If you can't be good, be weird