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SUMMARY:Hermann Karcher (University of Bonn)
DTSTART:20201215T180000Z
DTEND:20201215T190000Z
DTSTAMP:20260423T035019Z
UID:OSGA/41
DESCRIPTION:Title: Nu
merical experiments with closed constant curvature space curves\nby He
rmann Karcher (University of Bonn) as part of Online Seminar "Geometric An
alysis"\n\n\nAbstract\nThe discovery story will be told with pictures illu
strating all steps\, including the observation\nbelow which we us
ed but could not prove. Since the shape of space curves is difficult to be
\ncorrectly deduced from planar images\, most images will be red-green an
aglyphs. They\ncan be looked at without red-green glasses\, but without gi
ving the 3D impression.\n\nDec 15\,2020: H. Karcher: Closed consta
nt curvature space curves\n\nThe only closed constant curvature s
pace curves which I knew in 2004\nwere made from pieces of circles and hel
ices.\nThe Frenet equations allow to construct space curves of constant cu
rvature $\\kappa$\nby prescribing a torsion function $\\tau(s)$. For close
d curves one needs periodic\ntorsion functions\, for example Fourier polyn
omials. If one chooses these\nfunctions so that they are skew symmetric wi
th respect to their zeros\, $\\tau(a-s)=-\\tau(a+s)$\, then\nthe resulting
curves have the normal planes at these points as planes of\nmirror symmet
ry. If adjacent symmetry planes have angles such as $\\pi/3$\,\nthen the c
urves are forced by their symmetries to be closed. This gives the \nfirst
collection of new examples.\n\nIf the torsion functions are even with resp
ect to their extremal points\, i.e.\n$\\tau(a-s) = \\tau(a+s)$\, then the
resulting curves have their principal normals\nat these points as symmetry
axes for $180^\\circ$ rotations. If two such adjacent\nsymmetry normals a
re coplanar and intersect under rational angles
($\\pi p/q$)\,\nthen
the curves are again forced by their symmetries to close up. Therefore\non
e can hope to get examples by solving a 2-parameter problem.\n\nThis is ma
de simple by an observation which I cannot prove: The distance of
\nadjacent symmetry normals depends in a surprisingly monotone way on the
constant\nterm in the Fourier polynomial $\\tau(s) = b_0 + b_1\\\,\\sin(s
) + b_3\\\, \\sin(3s)$. This\nallows to consider $b_0 = b_0(\\kappa\, b_1\
, b_3)$\, such that the symmetry normals of the\nresulting curves are copl
anar and hence intersect all in one point. Therefore we\nhave again to sol
ve a 1-parameter problem by choosing $\\kappa\, b_1\, b_3$ in such a way\n
that adjacent symmetry normals intersect with a rational angle. This gives
a wealth\nof new examples.\n\nThe evolutes of such curves have also const
ant curvature $\\kappa$\, but they have \nsingularities at the zeros of $\
\tau(s)$. This led to a search for closed examples with $\\tau(s) > 0$.\nA
(2-11)-torus knot showed up and suggested to look for examples on tori. E
asy ones\nare again found by symmetries and more complicated ones as solut
ions of intermediate\nvalue problems. The formulas work also on cylinders
and revealed easier examples than\nall the previous ones!\n\nThen E. Tjade
n suggested to look for examples which are congruent to t
heir evolutes.\nThey were found by modifying the Frenet equations. The $(2
\, 2n+1)$- torus knots among\nthem are in fact their own evolutes.\n
LOCATION:https://researchseminars.org/talk/OSGA/41/
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