We propose a geometric construction of three-dimensional birational maps that preserve two pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along the straight line generators of the quadrics of the first pencil and are defined by the intersections with quadrics of the second pencil. On each quadric of the first pencil, the maps act as two-dimensional QRT maps. While these maps are of a pretty high degree in general, we find geometric conditions which guarantee that the degree is reduced to 3. The resulting degree 3 maps are illustrated by two known and two novel Kahan-type discretizations of three-dimensional Nambu systems, including the Euler top and the Zhukovski–Volterra gyrostat with two non-vanishing components of the gyrostatic momentum.

我们提出了三维双有理图的几何构造，保留了两支二次曲线。这些贴图充当对合的组合，而对合又沿着第一支铅笔的二次曲线的直线生成器起作用，并由与第二支铅笔的二次曲线的交点定义。在第一支铅笔的每个二次曲面上，贴图充当二维 QRT 贴图。虽然这些映射通常具有相当高的阶数，但我们发现保证阶数减少到 3 的几何条件。所得的阶数 3 映射由三维 Nambu 系统的两个已知和两个新颖的 Kahan 型离散化来说明，包括欧拉陀螺仪和朱可夫斯基-沃尔泰拉陀螺仪，其具有两个非零陀螺动量分量。