Re: Minimizing rate of change of curvature along a curve
- From: John Bailey <john_bailey@xxxxxxxxxxxxxxxx>
- Date: Wed, 24 Oct 2007 08:55:42 -0400
On 23 Oct 2007 22:51:37 -0400, "Keith F. Lynch" <kfl@xxxxxxxxxxxxxx>
wrote:
If you start at 0,1 heading in the +X direction and end at 1,0 headingAs noted by Robert Israel, your stated objective can be interpreted
in the -Y direction, along a curve that minimizes the rate of change
of curvature along its length, what kind of curve would you get?
If the initial and final curvature can be anything, the solution is
obviously a quarter circle centered on the origin, as the change of
curvature is zero everywhere. But what if the starting and ending
curvature are both zero? Then what kind of curve do you get? And
what's its arc length?
Thanks
in various ways. When I read your question, I immediately thought you
were asking about the railway spiral (aka the transition spiral). Such
a spiral is used on railroads to minimize the abrupt effect of
centrifugal force on occupants as the rail car turns.
..
http://mysite.du.edu/~jcalvert/railway/transpir.htm
"How such a spiral is determined mathematically is shown in the
Figure. (http://mysite.du.edu/~jcalvert/railway/transpir.gif) Taking
the origin at the T.S., the point where the tangent ends and the
spiral begins, the coordinates of a point on the spiral a distance s
from the T.S. are x and y. The angle chi is the angle of the tangent
to the spiral at (x,y). The first line shows that chi is proportional
to the square of the distance s from the T.S. Now we can express x and
y in terms of integrals, in fact taking chi as a parameter. These are
the Fresnel integrals familiar in physical optics, which are tabulated
and can be computed easily. They can also be expressed as integrals of
Bessel functions of order 1/2, and this can be worked into expressions
for x and y as infinite series of Bessel functions."
http://www.tfhrc.gov/pubrds/septoct01/spiral.htm
"The use of spirals was first documented in the late 1600s in "Sino
Loria," a treatise by James Bernouilli. Spirals were rediscovered in
1874 by Cornu and used in optics. Shortly afterward (sometime in the
1880s), spirals began to replace parabolic curves in easing
transitions for railroads. "
John
.
- References:
- Minimizing rate of change of curvature along a curve
- From: Keith F. Lynch
- Minimizing rate of change of curvature along a curve
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