In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in \({{\,\textrm{PG}\,}}(n,q)\) with q odd. We define relations on the anisotropic points of such a quadric that depend on the type of line spanned by the points and whether or not they are of the same “quadratic type”. This yields an imprimitive 5-class association scheme. We calculate the matrices of eigenvalues and dual eigenvalues of this scheme. We also use this result, together with similar results from the literature concerning other classical polar spaces, to exactly calculate the spectrum of orthogonality graphs on the anisotropic points of non-degenerate quadrics in odd characteristic and of non-degenerate Hermitian varieties. As a byproduct, we obtain a 3-class association scheme on the anisotropic points of non-degenerate Hermitian varieties, where the relation containing two points depends on the type of line spanned by these points, and whether or not they are orthogonal.

在本文中，我们研究了经典极空间各向异性点上的关联方案。我们的主要结果涉及 \（{{\，\textrm{PG}\，}}（n，q）\） 中具有 q 奇数的非简并椭圆和双曲二次函数。我们定义这种二次方的各向异性点上的关系，这取决于点跨越的线的类型以及它们是否属于相同的 “二次型”。这产生了一个不原始的 5 类关联方案。我们计算该方案的特征值和对偶特征值的矩阵。我们还使用这个结果，以及来自其他经典极空间的文献中的类似结果，来精确计算奇数特征中非简并二次方和非简并厄米特变体的各向异性点上的正交性图谱。作为副产品，我们在非简并埃尔米特品种的各向异性点上获得了 3 类关联方案，其中包含两个点的关系取决于这些点跨越的线的类型，以及它们是否是正交的。