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From: mike3 on 22 Jun 2010 21:26 Hi. Is it true that if one has a continuous curve connecting two points, makes a copy of it and slides the copy along the direction between those points a distance less than 1/2 the distance between the points, then the original curve and translated copy will intersect at at least one point? If not, can you provide me a counterexample, and if so, how about either a proof or just a hint at the proof, if the proof isn't too heavily sophisticated?
From: Robert Israel on 22 Jun 2010 22:03 mike3 <mike4ty4(a)yahoo.com> writes: > Hi. > > Is it true that if one has a continuous curve connecting two points, > makes a copy of it and slides the copy along the direction between > those points a distance less than 1/2 the distance between the points, > then the original curve and translated copy will intersect at at least > one point? If not, can you provide me a counterexample, and if so, how > about either a proof or just a hint at the proof, if the proof isn't > too heavily sophisticated? Counterexample: points are (0,0) and (1,1), curve is y = sin(9/2 pi x) - x for 0 <= x <= 1, slide right by distance 0.45. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Tim Little on 22 Jun 2010 22:11 On 2010-06-23, mike3 <mike4ty4(a)yahoo.com> wrote: > Is it true that if one has a continuous curve connecting two points, > makes a copy of it and slides the copy along the direction between > those points a distance less than 1/2 the distance between the points, > then the original curve and translated copy will intersect at at least > one point? No. > If not, can you provide me a counterexample A helix. - Tim
From: Robert Israel on 22 Jun 2010 22:21 > mike3 <mike4ty4(a)yahoo.com> writes: > > > Hi. > > > > Is it true that if one has a continuous curve connecting two points, > > makes a copy of it and slides the copy along the direction between > > those points a distance less than 1/2 the distance between the points, > > then the original curve and translated copy will intersect at at least > > one point? If not, can you provide me a counterexample, and if so, how > > about either a proof or just a hint at the proof, if the proof isn't > > too heavily sophisticated? > > Counterexample: points are (0,0) and (1,1), curve is y = sin(9/2 pi x) - x > for 0 <= x <= 1, slide right by distance 0.45. .... or even better, by distance 4/9. .... or more generally, for positive integer n, f(x) = sin((2n+1/2) pi x) - c, and slide right by distance 4/(4n+1). -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Tim Little on 22 Jun 2010 22:22
On 2010-06-23, Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote: > Counterexample: points are (0,0) and (1,1), curve is y = sin(9/2 pi x) - x > for 0 <= x <= 1, slide right by distance 0.45. Nice counterexample, though I suppose you really mean (0,0) and (1,0). - Tim |