We introduce new geometrical characteristics of any smooth probability distribution function $F$ on $\R^2$, such as bivariate quantiles and bivariate ranks. For this we build a two dimensional $r.v.$ generator $\mathcal{G}_F$ from the curves and curvilinear measures induced by what we call the Kendall ordering of $F$. The ensuing quantile transform $\tau_{FG}=\mathcal{G}_F\circ \mathcal{G}^{-1}_F$ of $F$ into $G$ is a closed form, continuous transport map having natural statistical applications. The fact that $\mathcal{G}_F$ and $\tau_{FG}$ are explicitly derived from $F$ makes their empirical counterparts $\mathcal{G}_{F,n}$ and $\tau_{n,m}$ fastly computable from samples of $F$ and $G$ of very large size $n$ and $m$. From the geometrical nature of $\mathcal{G}_{F,n}$ we construct non parametric depth parameters, local depth fields, contours and a bivariate Kendall tau. We also illustrate how $\tau_{n,m}$ can be used to quantify tests - goodness of fit, comparison, same copula or radiality. The paper focusses on basic properties of $\tau_{FG}$ then numerically illustrates most of the introduced notions.