4 research outputs found
Universal convex covering problems under translation and discrete rotations
We consider the smallest-area universal covering of planar objects of
perimeter 2 (or equivalently closed curves of length 2) allowing translation
and discrete rotations. In particular, we show that the solution is an
equilateral triangle of height 1 when translation and discrete rotation of
are allowed. Our proof is purely geometric and elementary. We also give
convex coverings of closed curves of length 2 under translation and discrete
rotations of multiples of and . We show a minimality of the
covering for discrete rotation of multiples of , which is an equilateral
triangle of height smaller than 1, and conjecture that the covering is the
smallest-area convex covering. Finally, we give the smallest-area convex
coverings of all unit segments under translation and discrete rotations
for all integers