Vector Fields for Mean Value Coordinates

Abstract

We find vector fields $\mathbf{F}$ that provide the representation $\mathbf{v}=\int_{\partial D}\mathbf{r}\,\mathbf{F}\cdot d\mathbf{r}/\int_{\partial D}\mathbf{F}\cdot d\mathbf{r}$ for any compact 2-D manifold $D\subset\mathbb{R}^2$ with piecewise smooth boundary $\partial D$ and $\mathbf{v}\in \mathbb{R}^2$, and the representation $\mathbf{v}=\iint_{\partial M}\mathbf{r}\,\mathbf{F}\cdot d\mathbf{S}/\iint_{\partial M}\mathbf{F}\cdot d\mathbf{S}$ for any compact 3-D manifold $M\subset\mathbb{R}^3$ and $\mathbf{v}\in\mathbb{R}^3$. Our method exploits properties of conservative fields in $\mathbb{R}^2$ and divergence free vector fields in $\mathbb{R}^3$. Discrete versions, which are more general than Floater's mean value coordinates, are derived from the above representations with a special choice of $\mathbf{F}$, either by taking points on the boundaries of $D\subset\mathbb{R}^2$ and $M\subset\mathbb{R}^3$ or by considering representations on boundaries of polygons in $\mathbb{R}^2$ or polyhedra in $\mathbb{R}^3$.