In $n$-dimensional projective space $n$-pararnetric family of $(n-2)$-dimensional non-degenerate quadrics (quadric elements) with given $(n-m-1)$-subspaces in their hyperplanes is considered. In § 1 the differential equations of the hypercomplex $V_n$ of quadric elements and fields of different geometrical objects on $V_n$ are given. In § 2 a distribution of $(n-m-1)$-subspaces in the hyperplanes of the hypercomplex $V_n$ is defined. Such a hypercomplex $V_n$ (hypercomplex $V_{n,m}$) is an $n$-dimensional manifold of pairs of figures $F_1$, $F_2$, where $F_1$ is a quadric element and $F_2$ — $(n-m-1)$-dimensional subspace of hyperplane of $F_1$. In § § 3, 4 the constructions are carried out in a polarcanonized moving frame. Different objects intrinsically defined by the tangent distribution are studied.