Ladnor Geissinger
Posts:
313
From:
University of North Carolina at Chapel Hill
Registered:
12/4/04


[mathlearn] Re: length and area; sidewalks and roads
Posted:
May 5, 2008 4:42 PM



Yes of course the actual curves on roads are not sections of circular rings, but could be thought of as the limit of a succession of narrow sections of such rings with gradually increasing and then decreasing curvature. The technical point is that such a region can be swept out by a line segment of length w with its center on the centerline of the road and always perpendicular to the centerline, because the radius of curvature of the centerline is always larger than w/2.
Note that the above method of sweeping out the road area does not work for the straightedged "sidewalks" described before, and even the notion of centerline is a bit tricky as you get near a sharp corner. But such details won't bother those just beginning to study geometry. Ladnor Geissinger
 In mathlearn@yahoogroups.com, "Ralph A. Raimi" <rarm@...> wrote: > > Ladnor Geissinger speaks of the areas of road surfaces, when they > have a turning somewhere, as if the curved portion has circular > boundaries. I believe that roadbuilders avoid circles here, since this > would imply a sudden transition from zero to a nonzero acceleration in > the normal direction, requiring a sudden jerk of the steering wheel. > Instead they make the curvature change more gradually from zero to the > maximum they intend, holding that max (circular motion) for a rather short > distance until they are ready for a gradual decline of curvature to > straighton motion. I have no idea what formulas they use, if any. It > might be a seatofthepants drawingboard solution, for all I know, for I > have noticed no uniformity in roadbuilding practice here, some turnings > being less comfortable than others in my rough experience. > > Ralph A. Raimi Tel. 585 275 4429 or (home) 585 244 9368 > Dept. of Mathematics, Univ.of Rochester, Rochester, NY 14627 > <http://www.math.rochester.edu/people/faculty/rarm/> > > "Algebra is conducive to symbolic reasoning." (PSSM, p.345) >

