English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the smallest-area universal covering of planar objects ofperimeter 2 (or equivalently closed curves of length 2) allowing translationand discrete rotations. In particular, we show that the solution is anequilateral triangle of height 1 when translation and discrete rotation of$\pi$ are allowed. Our proof is purely geometric and elementary. We also giveconvex coverings of closed curves of length 2 under translation and discreterotations of multiples of $\pi/2$ and $2\pi/3$. We show a minimality of thecovering for discrete rotation of multiples of $\pi/2$, which is an equilateraltriangle of height smaller than 1, and conjecture that the covering is thesmallest-area convex covering. Finally, we give the smallest-area convexcoverings of all unit segments under translation and discrete rotations$2\pi/k$ for all integers $k\ge 3$.

Universal convex covering problems under translation and discrete rotations