Energy and length in a topological planar quadrilateral

Cross Ratios of Quadrilateral Linkages

We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we describe the image of cross-ratio map for quadrilateral linkage and planar robot 3-arm.

四边形的一个几何性质

A geometric property of quadrilaterals

针对平面拟共形映射中的四边形，本文证明模处于 $[1/K,K]$ 的四边形内部必含一个半径由对边内部距离和 $K$ 决定的圆盘。

Reciprocal lower bound on modulus of curve families in metric surfaces

We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $ζ_1, ζ_2, ζ_3, ζ_4$ and let $Γ(ζ_i, ζ_j; Q)$ denote the family of curves in $Q$ connecting $ζ_i$ and $ζ_j$; then $\text{mod} Γ(ζ_1, ζ_3; Q) \text{mod} Γ(ζ_2, ζ_4; Q) \geq 1/κ$ for $κ= 2000^2\cdot (4/π)^2$. This answers a question concerning minimal hypotheses under which a metric spa…