Road design uses cubic spline curves, which have the properties that they
can be made to pass smoothly through any desired set of points, yet the
radius of curvature has no discontinuities. This means that you never have
to make a sudden change to the position of the steering wheel in order
to stay precisely in lane. The same is true of the catenary, which is actually
the curve formed by a dangling chain (with no stiffness). Bézier
curves are parametric cubics - they are to be found in some drawing and
CAD programs.
But I'm afraid I can think of no application of parametric cubics to
sundials. Sorry. Pity, because they are beautifully elegant.
Regards
Chris Lusby Taylor
51.4N, 1.3W
J
John Davis wrote:
Hi all, While
researching the BSS Glossary, I looked up the term "Lemniscate" in a mathematics
dictionary. This is the term used, I believe, for the analemma in
the Latin countries, and originally meant "ribbon-like". According
to the dictionary, in English the lemniscate curve is similar to a spiral
but differs from it principally because "... there is a slight falling
off of the rate of increase of radial acceleration as the distance from
the starting point increases" (!) As a result of this, it is used
in road design as a transition from a straight road into a circular arc. The
diagram in the dictionary comparing the lemniscate, spiral and cubic parabola
shows the first of these to look like one lobe of an analemma. So,
there was a closer connection between Mike's original question and dialling
than we might have originally thought! Best
regards, John-Dr
J R Davis
Flowton, UK
52.08N, 1.043E
email: [EMAIL PROTECTED]
- Original Message -
From:
Peter
Abrahams
To: sundial@rrz.uni-koeln.de
Sent: 11 February 2001 23:15
Subject: mystery curve
>There is a curve for which the transition from one radius of curvature
to
>another is as gradual as possible. I too have forgotten its
name however it
>can be formed by bending a length of material of constant stiffness
(such as
>a garden hose) into a loop by holding only its ends.
That's a catenary.
>I don't think the occupants of a car would guide it through a path
that
>would cause them discomfort. So even if a road did have a jump
in radius of
>curvature, the path traced out by a car wouldn't. (Unlike a
roller coaster,
>a car doesn't have to travel out any particular path).
No, but you do want to encourage drivers to stay in their lanes.
--Peter
___
Peter Abrahams [EMAIL PROTECTED]
The history of the telescope &
the binocular: http://www.europa.com/~telscope/binotele.htm