On the preserving of the orientation of triangle by quasi-isometric mapping

The present paper proposes quantitative characteristics the two triangle contiguity ratio, which is the distance $\rho $ in the space of $4$-point families from the family $ X $, defined by a given pair of adjacent triangles, up to a set $\mathcal{Y}^*$ families defined by all kinds of pairs non-adjacent triangles (with common party). This characteristic is local sufficient sign of no grid overflow at quasi-isometric mapping and can be applied to making triangulation of a given region as an image of some a reference triangulated region. $\rho(X, \mathcal{Y} ^ *)$ is required to calculate structurally specify in $\mathcal{Y}^*$ some subset, distance from $X$ to which is $\rho(X, \mathcal{Y}^*)$. This requires, in turn, splitting the set $\mathcal{Y}^*$ into $15$ classes and studies of each of them for exclusion “extra” families and descriptions of the remaining ones. Due to the large volume of full study in the article only three classes of these $15$ are examined. Two of them are “nodal” in the general study scheme, by example the third shows the combinatorial nature of the task.